| L(s) = 1 | + 2-s + 3-s − 2·5-s + 6-s + 2·7-s − 8-s − 2·10-s − 2·11-s + 5·13-s + 2·14-s − 2·15-s − 16-s − 5·17-s + 2·19-s + 2·21-s − 2·22-s − 6·23-s − 24-s − 7·25-s + 5·26-s − 27-s + 9·29-s − 2·30-s − 8·31-s − 2·33-s − 5·34-s − 4·35-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 0.894·5-s + 0.408·6-s + 0.755·7-s − 0.353·8-s − 0.632·10-s − 0.603·11-s + 1.38·13-s + 0.534·14-s − 0.516·15-s − 1/4·16-s − 1.21·17-s + 0.458·19-s + 0.436·21-s − 0.426·22-s − 1.25·23-s − 0.204·24-s − 7/5·25-s + 0.980·26-s − 0.192·27-s + 1.67·29-s − 0.365·30-s − 1.43·31-s − 0.348·33-s − 0.857·34-s − 0.676·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.237937436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.237937436\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 13 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 5 T + 8 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 11 T + 60 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 14 T + 125 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 2 T - 85 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−14.72335395250540385418313040984, −14.02284794869150380654077828001, −13.74556078736786250052476462286, −13.22864199670998726249811629288, −12.78797333273494567495868239787, −11.87481719325640869891287111228, −11.50830975629029436416191828239, −11.24536489118274434866906356977, −10.32309361018841613699534679725, −9.813277683056482657183578413109, −8.651546379069765902010011758834, −8.606747222634992294453947391565, −7.892840979726185351712776661992, −7.35495711273524785367998704520, −6.31853884050067719990124591716, −5.71353738541044452852246427228, −4.74671246253920085262181588315, −4.06164971991764944799633028143, −3.49199774966556626951947285055, −2.19914849365351686454379623697,
2.19914849365351686454379623697, 3.49199774966556626951947285055, 4.06164971991764944799633028143, 4.74671246253920085262181588315, 5.71353738541044452852246427228, 6.31853884050067719990124591716, 7.35495711273524785367998704520, 7.892840979726185351712776661992, 8.606747222634992294453947391565, 8.651546379069765902010011758834, 9.813277683056482657183578413109, 10.32309361018841613699534679725, 11.24536489118274434866906356977, 11.50830975629029436416191828239, 11.87481719325640869891287111228, 12.78797333273494567495868239787, 13.22864199670998726249811629288, 13.74556078736786250052476462286, 14.02284794869150380654077828001, 14.72335395250540385418313040984
