L(s) = 1 | + 2·2-s + 2·4-s + 5·7-s + 4·8-s − 12·11-s + 2·13-s + 10·14-s + 5·16-s + 11·17-s + 16·19-s − 24·22-s + 7·23-s + 4·26-s + 10·28-s + 29-s + 5·31-s + 22·34-s + 17·37-s + 32·38-s − 19·41-s + 14·43-s − 24·44-s + 14·46-s − 6·47-s + 3·49-s + 4·52-s − 15·53-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 4-s + 1.88·7-s + 1.41·8-s − 3.61·11-s + 0.554·13-s + 2.67·14-s + 5/4·16-s + 2.66·17-s + 3.67·19-s − 5.11·22-s + 1.45·23-s + 0.784·26-s + 1.88·28-s + 0.185·29-s + 0.898·31-s + 3.77·34-s + 2.79·37-s + 5.19·38-s − 2.96·41-s + 2.13·43-s − 3.61·44-s + 2.06·46-s − 0.875·47-s + 3/7·49-s + 0.554·52-s − 2.06·53-s + ⋯ |
Λ(s)=(=((314⋅514⋅237)s/2ΓC(s)7L(s)Λ(2−s)
Λ(s)=(=((314⋅514⋅237)s/2ΓC(s+1/2)7L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
43.58523449 |
L(21) |
≈ |
43.58523449 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 5 | 1 |
| 23 | (1−T)7 |
good | 2 | 1−pT+pT2−p2T3+7T4−p2T5+pT6−3pT7+p2T8−p4T9+7p3T10−p6T11+p6T12−p7T13+p7T14 |
| 7 | 1−5T+22T2−69T3+242T4−795T5+2389T6−6782T7+2389pT8−795p2T9+242p3T10−69p4T11+22p5T12−5p6T13+p7T14 |
| 11 | 1+12T+85T2+454T3+2117T4+8636T5+31593T6+106948T7+31593pT8+8636p2T9+2117p3T10+454p4T11+85p5T12+12p6T13+p7T14 |
| 13 | 1−2T+5pT2−138T3+2003T4−4230T5+38379T6−71996T7+38379pT8−4230p2T9+2003p3T10−138p4T11+5p6T12−2p6T13+p7T14 |
| 17 | 1−11T+118T2−847T3+5650T4−1797pT5+154799T6−657626T7+154799pT8−1797p3T9+5650p3T10−847p4T11+118p5T12−11p6T13+p7T14 |
| 19 | 1−16T+215T2−1878T3+14471T4−87496T5+480449T6−2186420T7+480449pT8−87496p2T9+14471p3T10−1878p4T11+215p5T12−16p6T13+p7T14 |
| 29 | 1−T+128T2−245T3+8366T4−17091T5+357125T6−637966T7+357125pT8−17091p2T9+8366p3T10−245p4T11+128p5T12−p6T13+p7T14 |
| 31 | 1−5T+136T2−741T3+9446T4−47311T5+432575T6−1806862T7+432575pT8−47311p2T9+9446p3T10−741p4T11+136p5T12−5p6T13+p7T14 |
| 37 | 1−17T+326T2−3777T3+40884T4−349491T5+2661449T6−17243126T7+2661449pT8−349491p2T9+40884p3T10−3777p4T11+326p5T12−17p6T13+p7T14 |
| 41 | 1+19T+394T2+4701T3+55916T4+481409T5+4052625T6+26296414T7+4052625pT8+481409p2T9+55916p3T10+4701p4T11+394p5T12+19p6T13+p7T14 |
| 43 | 1−14T+225T2−2344T3+23113T4−197474T5+1484753T6−10496528T7+1484753pT8−197474p2T9+23113p3T10−2344p4T11+225p5T12−14p6T13+p7T14 |
| 47 | 1+6T+113T2+766T3+7517T4+50706T5+433805T6+2517332T7+433805pT8+50706p2T9+7517p3T10+766p4T11+113p5T12+6p6T13+p7T14 |
| 53 | 1+15T+208T2+1457T3+14012T4+92953T5+981327T6+6221462T7+981327pT8+92953p2T9+14012p3T10+1457p4T11+208p5T12+15p6T13+p7T14 |
| 59 | 1+11T+230T2+1599T3+18416T4+66169T5+690007T6+1175722T7+690007pT8+66169p2T9+18416p3T10+1599p4T11+230p5T12+11p6T13+p7T14 |
| 61 | 1−8T+145T2−942T3+15371T4−104880T5+1138851T6−6201236T7+1138851pT8−104880p2T9+15371p3T10−942p4T11+145p5T12−8p6T13+p7T14 |
| 67 | 1+3T+118T2+155T3+8162T4+19373T5+678793T6+2355010T7+678793pT8+19373p2T9+8162p3T10+155p4T11+118p5T12+3p6T13+p7T14 |
| 71 | 1−T+232T2+193T3+30842T4+36365T5+3021963T6+3211222T7+3021963pT8+36365p2T9+30842p3T10+193p4T11+232p5T12−p6T13+p7T14 |
| 73 | 1−18T+493T2−6482T3+103019T4−1050182T5+12177991T6−98243980T7+12177991pT8−1050182p2T9+103019p3T10−6482p4T11+493p5T12−18p6T13+p7T14 |
| 79 | 1+2T+265T2−12T3+29517T4−114626T5+2111557T6−15982568T7+2111557pT8−114626p2T9+29517p3T10−12p4T11+265p5T12+2p6T13+p7T14 |
| 83 | 1+T+346T2+1005T3+59632T4+224715T5+6904107T6+24596814T7+6904107pT8+224715p2T9+59632p3T10+1005p4T11+346p5T12+p6T13+p7T14 |
| 89 | 1+16T+455T2+5800T3+94517T4+1025952T5+12462835T6+113567824T7+12462835pT8+1025952p2T9+94517p3T10+5800p4T11+455p5T12+16p6T13+p7T14 |
| 97 | 1−26T+727T2−12432T3+210405T4−2712758T5+33692675T6−338833216T7+33692675pT8−2712758p2T9+210405p3T10−12432p4T11+727p5T12−26p6T13+p7T14 |
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L(s)=p∏ j=1∏14(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−3.67622770805796589384485524624, −3.56469555456623493417008919666, −3.45716728962821934041412097839, −3.28995321715952124922929882832, −3.23262147138743849076422661964, −3.16699766154011162100316583276, −2.99330207757845276578569083605, −2.96362940323382753446566629888, −2.85971139080819872327909040418, −2.58322594850782075950632283470, −2.44652476204408738712624434938, −2.38344957877817988169252468426, −2.26870317908304011734157154213, −2.25880784801975331959579619232, −1.74875604923383414949773294600, −1.69201810164252663486875508493, −1.59181251169268922075121550421, −1.58807710115916272239700248441, −1.38393619385922293234686809489, −1.12755690715627160119586567899, −0.970801217109611142879721505640, −0.73997415199205457410465997546, −0.72491032411176141644869221386, −0.62605450261852490887545447240, −0.20894264582800039865576681169,
0.20894264582800039865576681169, 0.62605450261852490887545447240, 0.72491032411176141644869221386, 0.73997415199205457410465997546, 0.970801217109611142879721505640, 1.12755690715627160119586567899, 1.38393619385922293234686809489, 1.58807710115916272239700248441, 1.59181251169268922075121550421, 1.69201810164252663486875508493, 1.74875604923383414949773294600, 2.25880784801975331959579619232, 2.26870317908304011734157154213, 2.38344957877817988169252468426, 2.44652476204408738712624434938, 2.58322594850782075950632283470, 2.85971139080819872327909040418, 2.96362940323382753446566629888, 2.99330207757845276578569083605, 3.16699766154011162100316583276, 3.23262147138743849076422661964, 3.28995321715952124922929882832, 3.45716728962821934041412097839, 3.56469555456623493417008919666, 3.67622770805796589384485524624
Plot not available for L-functions of degree greater than 10.