| L(s) = 1 | + 54·2-s − 120·3-s + 1.46e3·4-s − 6.48e3·6-s − 3.36e4·7-s + 1.10e5·8-s − 2.22e5·9-s − 7.50e5·11-s − 1.75e5·12-s + 9.54e3·13-s − 1.81e6·14-s + 5.63e6·16-s − 4.16e6·17-s − 1.19e7·18-s − 1.79e7·19-s + 4.03e6·21-s − 4.05e7·22-s + 6.61e7·23-s − 1.32e7·24-s + 5.15e5·26-s + 3.38e7·27-s − 4.90e7·28-s + 6.15e7·29-s − 1.52e7·31-s + 7.79e7·32-s + 9.00e7·33-s − 2.24e8·34-s + ⋯ |
| L(s) = 1 | + 1.19·2-s − 0.285·3-s + 0.712·4-s − 0.340·6-s − 0.755·7-s + 1.19·8-s − 1.25·9-s − 1.40·11-s − 0.203·12-s + 0.00713·13-s − 0.902·14-s + 1.34·16-s − 0.710·17-s − 1.49·18-s − 1.66·19-s + 0.215·21-s − 1.67·22-s + 2.14·23-s − 0.340·24-s + 0.00851·26-s + 0.453·27-s − 0.538·28-s + 0.556·29-s − 0.0958·31-s + 0.410·32-s + 0.400·33-s − 0.847·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30625 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.767473472\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.767473472\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p^{5} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - 27 p T + 91 p^{4} T^{2} - 27 p^{12} T^{3} + p^{22} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + 40 p T + 26290 p^{2} T^{2} + 40 p^{12} T^{3} + p^{22} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 68256 p T + 653251941286 T^{2} + 68256 p^{12} T^{3} + p^{22} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 9548 T + 580074739914 T^{2} - 9548 p^{11} T^{3} + p^{22} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4160052 T + 72837924685942 T^{2} + 4160052 p^{11} T^{3} + p^{22} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 17998712 T + 293399338219938 T^{2} + 17998712 p^{11} T^{3} + p^{22} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 66161016 T + 2994683043115918 T^{2} - 66161016 p^{11} T^{3} + p^{22} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2121228 p T + 12823193731307518 T^{2} - 2121228 p^{12} T^{3} + p^{22} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 15281552 T + 44544265736191854 T^{2} + 15281552 p^{11} T^{3} + p^{22} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 527218340 T + 315935809764623790 T^{2} - 527218340 p^{11} T^{3} + p^{22} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 178276140 T + 1103495454485680198 T^{2} + 178276140 p^{11} T^{3} + p^{22} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 1826745232 T + 2408495597390567334 T^{2} + 1826745232 p^{11} T^{3} + p^{22} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 568240704 T + 125222806434661774 T^{2} + 568240704 p^{11} T^{3} + p^{22} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4185816372 T + 8708326678160294206 T^{2} - 4185816372 p^{11} T^{3} + p^{22} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 3111345000 T + 25961868574953911218 T^{2} - 3111345000 p^{11} T^{3} + p^{22} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 15042595060 T + \)\(13\!\cdots\!78\)\( T^{2} - 15042595060 p^{11} T^{3} + p^{22} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 9856523968 T + \)\(12\!\cdots\!18\)\( T^{2} + 9856523968 p^{11} T^{3} + p^{22} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 24312011328 T + \)\(59\!\cdots\!02\)\( T^{2} + 24312011328 p^{11} T^{3} + p^{22} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 30890001932 T + \)\(73\!\cdots\!46\)\( T^{2} - 30890001932 p^{11} T^{3} + p^{22} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1992804256 T + \)\(85\!\cdots\!38\)\( T^{2} - 1992804256 p^{11} T^{3} + p^{22} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 5277014568 T + \)\(22\!\cdots\!46\)\( T^{2} + 5277014568 p^{11} T^{3} + p^{22} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 101541312828 T + \)\(76\!\cdots\!78\)\( T^{2} + 101541312828 p^{11} T^{3} + p^{22} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 192228621116 T + \)\(23\!\cdots\!14\)\( T^{2} - 192228621116 p^{11} T^{3} + p^{22} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.99533799497251596088279201993, −10.44872300198661882941095791710, −10.35255125457958396811050330374, −9.481925143875643634925665676721, −8.766802917438895353368923990851, −8.406578965395051766850139195228, −7.934841566244686388827528364128, −6.91809026701870038146392150993, −6.91295602763620197322175792995, −6.19101338989329804896488319089, −5.40138793371749692029644439859, −5.36858776407250199398693416871, −4.58423438782076973348576696105, −4.27323797046053642523126380734, −3.36250107873006060477389375345, −2.97097924926627517587399364779, −2.41558585353040226758569586394, −1.91911949183553563809916743772, −0.821408929906589560545650552818, −0.33702950726814817165123946365,
0.33702950726814817165123946365, 0.821408929906589560545650552818, 1.91911949183553563809916743772, 2.41558585353040226758569586394, 2.97097924926627517587399364779, 3.36250107873006060477389375345, 4.27323797046053642523126380734, 4.58423438782076973348576696105, 5.36858776407250199398693416871, 5.40138793371749692029644439859, 6.19101338989329804896488319089, 6.91295602763620197322175792995, 6.91809026701870038146392150993, 7.934841566244686388827528364128, 8.406578965395051766850139195228, 8.766802917438895353368923990851, 9.481925143875643634925665676721, 10.35255125457958396811050330374, 10.44872300198661882941095791710, 10.99533799497251596088279201993
