| L(s) = 1 | + 4·3-s − 5·5-s + 27·9-s − 68·11-s + 44·13-s − 20·15-s + 30·17-s − 108·19-s − 184·23-s + 260·27-s + 332·29-s + 32·31-s − 272·33-s + 370·37-s + 176·39-s + 308·41-s + 424·43-s − 135·45-s + 512·47-s + 120·51-s + 98·53-s + 340·55-s − 432·57-s + 860·59-s − 390·61-s − 220·65-s − 60·67-s + ⋯ |
| L(s) = 1 | + 0.769·3-s − 0.447·5-s + 9-s − 1.86·11-s + 0.938·13-s − 0.344·15-s + 0.428·17-s − 1.30·19-s − 1.66·23-s + 1.85·27-s + 2.12·29-s + 0.185·31-s − 1.43·33-s + 1.64·37-s + 0.722·39-s + 1.17·41-s + 1.50·43-s − 0.447·45-s + 1.58·47-s + 0.329·51-s + 0.253·53-s + 0.833·55-s − 1.00·57-s + 1.89·59-s − 0.818·61-s − 0.419·65-s − 0.109·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.561463231\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.561463231\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T - 11 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 68 T + 3293 T^{2} + 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 22 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 30 T - 4013 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 108 T + 4805 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 8 p T + 41 p^{2} T^{2} + 8 p^{4} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 166 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 32 T - 28767 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 11 p T + p^{3} T^{2} )( 1 + p T + p^{3} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 154 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 212 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 512 T + 158321 T^{2} - 512 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 98 T - 139273 T^{2} - 98 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 860 T + 534221 T^{2} - 860 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 390 T - 74881 T^{2} + 390 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 60 T - 297163 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 840 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 630 T + 7883 T^{2} - 630 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 1312 T + 1228305 T^{2} + 1312 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 436 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 598 T - 347365 T^{2} - 598 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 914 T + p^{3} T^{2} )^{2} \) |
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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
−9.882306451581767618605726674103, −9.491901781710410759330073863899, −8.781314524375429385499373759450, −8.395452817497608972111264161647, −8.259475224556903543834079244397, −7.898688753019068858832134520901, −7.37351674279971008136246211115, −7.08401304060087931577182455957, −6.28936984734017418800685127846, −6.07437199303658240103494979385, −5.61328980459960451543984251851, −4.77413922158320375315021075362, −4.50616383823537500964966803622, −4.03950702249381331749965026588, −3.60764411981358839231563436616, −2.72454594666112782450878312589, −2.54706486296859878897464588952, −2.05079902847655066686134932354, −0.875598635742120730867206931099, −0.67573247501840796191388402387,
0.67573247501840796191388402387, 0.875598635742120730867206931099, 2.05079902847655066686134932354, 2.54706486296859878897464588952, 2.72454594666112782450878312589, 3.60764411981358839231563436616, 4.03950702249381331749965026588, 4.50616383823537500964966803622, 4.77413922158320375315021075362, 5.61328980459960451543984251851, 6.07437199303658240103494979385, 6.28936984734017418800685127846, 7.08401304060087931577182455957, 7.37351674279971008136246211115, 7.898688753019068858832134520901, 8.259475224556903543834079244397, 8.395452817497608972111264161647, 8.781314524375429385499373759450, 9.491901781710410759330073863899, 9.882306451581767618605726674103
