| L(s) = 1 | − 2-s − 3-s + 4-s + 2·5-s + 6-s − 8-s + 9-s − 2·10-s − 12-s − 2·15-s + 16-s − 18-s + 6·19-s + 2·20-s − 18·23-s + 24-s − 2·25-s − 27-s + 5·29-s + 2·30-s − 32-s + 36-s − 6·38-s − 2·40-s − 8·43-s + 2·45-s + 18·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.288·12-s − 0.516·15-s + 1/4·16-s − 0.235·18-s + 1.37·19-s + 0.447·20-s − 3.75·23-s + 0.204·24-s − 2/5·25-s − 0.192·27-s + 0.928·29-s + 0.365·30-s − 0.176·32-s + 1/6·36-s − 0.973·38-s − 0.316·40-s − 1.21·43-s + 0.298·45-s + 2.65·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 501120 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 501120 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - T + p T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 6 T + p T^{2} ) \) |
| good | 7 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 23 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 168 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + T + p T^{2} ) \) |
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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
−8.230100141657985557849489435250, −7.956565732189732468315412547789, −7.46403945597617399803950970892, −6.78634285677377178315975134243, −6.51016260978630054762316820903, −5.93713530029459233116848294717, −5.67540219206073594222014331118, −5.19528798876182961985881819660, −4.52119969491088226755422172603, −3.83472772392576285729652994734, −3.37631124663471782883952957234, −2.35985783166777292597201677217, −1.98467852442659433375600678338, −1.23027457691363456138582337458, 0,
1.23027457691363456138582337458, 1.98467852442659433375600678338, 2.35985783166777292597201677217, 3.37631124663471782883952957234, 3.83472772392576285729652994734, 4.52119969491088226755422172603, 5.19528798876182961985881819660, 5.67540219206073594222014331118, 5.93713530029459233116848294717, 6.51016260978630054762316820903, 6.78634285677377178315975134243, 7.46403945597617399803950970892, 7.956565732189732468315412547789, 8.230100141657985557849489435250
