| L(s) = 1 | + 3-s + 2·5-s + 9-s − 13-s + 2·15-s + 17-s − 25-s + 27-s + 2·29-s + 8·31-s − 10·37-s − 39-s − 6·41-s + 4·43-s + 2·45-s − 8·47-s − 7·49-s + 51-s + 2·53-s + 4·59-s + 10·61-s − 2·65-s − 8·67-s − 12·71-s + 6·73-s − 75-s − 8·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s − 0.277·13-s + 0.516·15-s + 0.242·17-s − 1/5·25-s + 0.192·27-s + 0.371·29-s + 1.43·31-s − 1.64·37-s − 0.160·39-s − 0.937·41-s + 0.609·43-s + 0.298·45-s − 1.16·47-s − 49-s + 0.140·51-s + 0.274·53-s + 0.520·59-s + 1.28·61-s − 0.248·65-s − 0.977·67-s − 1.42·71-s + 0.702·73-s − 0.115·75-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−14.89563666194134, −14.27398791026170, −14.03680958775828, −13.48230492636872, −13.00742657421338, −12.54350536350446, −11.70179460832524, −11.59467505530022, −10.46769795093012, −10.20466227828672, −9.807324761294734, −9.187735064080770, −8.607295296614577, −8.177076228726735, −7.543310990032287, −6.788994823987076, −6.492094593629314, −5.669684274566616, −5.190735574583293, −4.534263666124136, −3.830676390052303, −3.068259193193784, −2.577312732730507, −1.784271816594680, −1.255217248343291, 0,
1.255217248343291, 1.784271816594680, 2.577312732730507, 3.068259193193784, 3.830676390052303, 4.534263666124136, 5.190735574583293, 5.669684274566616, 6.492094593629314, 6.788994823987076, 7.543310990032287, 8.177076228726735, 8.607295296614577, 9.187735064080770, 9.807324761294734, 10.20466227828672, 10.46769795093012, 11.59467505530022, 11.70179460832524, 12.54350536350446, 13.00742657421338, 13.48230492636872, 14.03680958775828, 14.27398791026170, 14.89563666194134
