| L(s) = 1 | + 2·3-s − 5-s + 9-s − 3·11-s + 6·13-s − 2·15-s − 2·17-s − 19-s − 23-s − 4·25-s − 4·27-s + 8·29-s − 8·31-s − 6·33-s + 6·37-s + 12·39-s − 2·41-s − 13·43-s − 45-s − 13·47-s − 4·51-s + 8·53-s + 3·55-s − 2·57-s + 12·59-s − 5·61-s − 6·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s + 1/3·9-s − 0.904·11-s + 1.66·13-s − 0.516·15-s − 0.485·17-s − 0.229·19-s − 0.208·23-s − 4/5·25-s − 0.769·27-s + 1.48·29-s − 1.43·31-s − 1.04·33-s + 0.986·37-s + 1.92·39-s − 0.312·41-s − 1.98·43-s − 0.149·45-s − 1.89·47-s − 0.560·51-s + 1.09·53-s + 0.404·55-s − 0.264·57-s + 1.56·59-s − 0.640·61-s − 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 13 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.83441847256653110965162081129, −6.96738712642747563196289872391, −6.21583437854160514040072675088, −5.45949337947767559251868514386, −4.50590620694394873552237989235, −3.71653224419339898837647615357, −3.22498796655634757404780894761, −2.36243648952996462728576523099, −1.49225369925413203882010646180, 0,
1.49225369925413203882010646180, 2.36243648952996462728576523099, 3.22498796655634757404780894761, 3.71653224419339898837647615357, 4.50590620694394873552237989235, 5.45949337947767559251868514386, 6.21583437854160514040072675088, 6.96738712642747563196289872391, 7.83441847256653110965162081129
