| L(s) = 1 | − 3-s − 2·5-s + 7-s + 9-s + 2·11-s + 2·13-s + 2·15-s + 2·17-s − 19-s − 21-s − 6·23-s − 25-s − 27-s − 8·31-s − 2·33-s − 2·35-s + 6·37-s − 2·39-s − 8·43-s − 2·45-s + 6·47-s + 49-s − 2·51-s − 4·55-s + 57-s + 10·61-s + 63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.603·11-s + 0.554·13-s + 0.516·15-s + 0.485·17-s − 0.229·19-s − 0.218·21-s − 1.25·23-s − 1/5·25-s − 0.192·27-s − 1.43·31-s − 0.348·33-s − 0.338·35-s + 0.986·37-s − 0.320·39-s − 1.21·43-s − 0.298·45-s + 0.875·47-s + 1/7·49-s − 0.280·51-s − 0.539·55-s + 0.132·57-s + 1.28·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 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
−7.67445696965532786631253236409, −7.03979823144996245772390244974, −6.17452434797620887172283868847, −5.64254124598763688851351878162, −4.72388304720903810316256764381, −3.95013189578563926890851419670, −3.55122469782048680114524293870, −2.16011565678599307668047908354, −1.18502392684665963586767921318, 0,
1.18502392684665963586767921318, 2.16011565678599307668047908354, 3.55122469782048680114524293870, 3.95013189578563926890851419670, 4.72388304720903810316256764381, 5.64254124598763688851351878162, 6.17452434797620887172283868847, 7.03979823144996245772390244974, 7.67445696965532786631253236409
