| L(s) = 1 | + 2-s + 3-s + 6-s + 2·7-s − 8-s + 3·11-s − 5·13-s + 2·14-s − 16-s − 8·19-s + 2·21-s + 3·22-s + 9·23-s − 24-s − 5·26-s − 27-s − 6·29-s − 8·31-s + 3·33-s + 11·37-s − 8·38-s − 5·39-s + 2·42-s − 10·43-s + 9·46-s + 24·47-s − 48-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 0.904·11-s − 1.38·13-s + 0.534·14-s − 1/4·16-s − 1.83·19-s + 0.436·21-s + 0.639·22-s + 1.87·23-s − 0.204·24-s − 0.980·26-s − 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.522·33-s + 1.80·37-s − 1.29·38-s − 0.800·39-s + 0.308·42-s − 1.52·43-s + 1.32·46-s + 3.50·47-s − 0.144·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.503189126\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.503189126\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 8 T - 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 9 T + 10 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 12 T + 55 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 17 T + 192 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
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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
−9.618750351952307851728575512748, −9.046345653021531231169721154200, −8.604605967231026929795067834896, −8.341453283788968935515330372207, −7.64762953222137301232560220090, −7.47997880128991289909167823833, −7.09359334407287365067668298456, −6.58247967296796472023918155032, −6.26155422779745777276167966699, −5.51475907805214680324697833555, −5.46391513641195370398040686551, −4.74264311305001154484679152704, −4.54244474376348163050864179933, −3.97492176273619082456330805102, −3.78326524009231916515257495863, −2.97619297213863808549794826993, −2.61074573397901374117442782629, −2.06576007088424769813982653234, −1.57197668024289001089001731601, −0.56909467726636134417425067103,
0.56909467726636134417425067103, 1.57197668024289001089001731601, 2.06576007088424769813982653234, 2.61074573397901374117442782629, 2.97619297213863808549794826993, 3.78326524009231916515257495863, 3.97492176273619082456330805102, 4.54244474376348163050864179933, 4.74264311305001154484679152704, 5.46391513641195370398040686551, 5.51475907805214680324697833555, 6.26155422779745777276167966699, 6.58247967296796472023918155032, 7.09359334407287365067668298456, 7.47997880128991289909167823833, 7.64762953222137301232560220090, 8.341453283788968935515330372207, 8.604605967231026929795067834896, 9.046345653021531231169721154200, 9.618750351952307851728575512748
