| L(s) = 1 | + 3-s + 5-s − 3·7-s + 9-s − 13-s + 15-s + 2·17-s + 19-s − 3·21-s − 4·23-s + 25-s + 27-s − 2·29-s − 31-s − 3·35-s − 3·37-s − 39-s − 4·41-s − 4·43-s + 45-s − 6·47-s + 2·49-s + 2·51-s + 57-s − 2·59-s − 61-s − 3·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.13·7-s + 1/3·9-s − 0.277·13-s + 0.258·15-s + 0.485·17-s + 0.229·19-s − 0.654·21-s − 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.179·31-s − 0.507·35-s − 0.493·37-s − 0.160·39-s − 0.624·41-s − 0.609·43-s + 0.149·45-s − 0.875·47-s + 2/7·49-s + 0.280·51-s + 0.132·57-s − 0.260·59-s − 0.128·61-s − 0.377·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.36701779765924, −13.01248244848257, −12.47659586203388, −11.95856070590039, −11.70673832485206, −10.79035891736048, −10.41477230606647, −9.933001623904554, −9.626674895547167, −9.141485661207442, −8.715708690250756, −8.072717413254692, −7.607824756588554, −7.140721526195168, −6.492441955863978, −6.210116618484789, −5.602439916946923, −5.015457088915666, −4.471490902260289, −3.716804847097553, −3.247094931480294, −2.984466517745330, −2.015891589841505, −1.793604375027535, −0.7792808378607277, 0,
0.7792808378607277, 1.793604375027535, 2.015891589841505, 2.984466517745330, 3.247094931480294, 3.716804847097553, 4.471490902260289, 5.015457088915666, 5.602439916946923, 6.210116618484789, 6.492441955863978, 7.140721526195168, 7.607824756588554, 8.072717413254692, 8.715708690250756, 9.141485661207442, 9.626674895547167, 9.933001623904554, 10.41477230606647, 10.79035891736048, 11.70673832485206, 11.95856070590039, 12.47659586203388, 13.01248244848257, 13.36701779765924
