L(s) = 1 | − 3·4-s − 2·5-s + 9-s + 2·11-s + 5·16-s + 6·20-s − 25-s − 12·29-s − 16·31-s − 3·36-s − 4·41-s − 6·44-s − 2·45-s + 2·49-s − 4·55-s − 8·59-s + 12·61-s − 3·64-s − 8·79-s − 10·80-s + 81-s − 12·89-s + 2·99-s + 3·100-s + 4·101-s − 4·109-s + 36·116-s + ⋯ |
L(s) = 1 | − 3/2·4-s − 0.894·5-s + 1/3·9-s + 0.603·11-s + 5/4·16-s + 1.34·20-s − 1/5·25-s − 2.22·29-s − 2.87·31-s − 1/2·36-s − 0.624·41-s − 0.904·44-s − 0.298·45-s + 2/7·49-s − 0.539·55-s − 1.04·59-s + 1.53·61-s − 3/8·64-s − 0.900·79-s − 1.11·80-s + 1/9·81-s − 1.27·89-s + 0.201·99-s + 3/10·100-s + 0.398·101-s − 0.383·109-s + 3.34·116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−10.17772545901372850796013638635, −9.705159010656125332480920056237, −9.083856298436522884927706897826, −8.975261543279013968212899152873, −8.281293500899591134516470178784, −7.46995816184461358823952634116, −7.43398897048000830563917400761, −6.48836008647392510605014123287, −5.45807303096955976252785458823, −5.31622482563829752699303340190, −4.20407903283329927801590362766, −3.95949329332453626615268855815, −3.41592277941264934062420800268, −1.76559017524548890655133359279, 0,
1.76559017524548890655133359279, 3.41592277941264934062420800268, 3.95949329332453626615268855815, 4.20407903283329927801590362766, 5.31622482563829752699303340190, 5.45807303096955976252785458823, 6.48836008647392510605014123287, 7.43398897048000830563917400761, 7.46995816184461358823952634116, 8.281293500899591134516470178784, 8.975261543279013968212899152873, 9.083856298436522884927706897826, 9.705159010656125332480920056237, 10.17772545901372850796013638635