L(s) = 1 | + 7·3-s + 15·5-s − 18·7-s + 22·9-s − 27·11-s + 57·13-s + 105·15-s − 44·17-s − 152·19-s − 126·21-s − 152·23-s + 100·25-s − 35·27-s + 29·29-s − 173·31-s − 189·33-s − 270·35-s + 120·37-s + 399·39-s − 314·41-s − 339·43-s + 330·45-s − 357·47-s − 19·49-s − 308·51-s + 59·53-s − 405·55-s + ⋯ |
L(s) = 1 | + 1.34·3-s + 1.34·5-s − 0.971·7-s + 0.814·9-s − 0.740·11-s + 1.21·13-s + 1.80·15-s − 0.627·17-s − 1.83·19-s − 1.30·21-s − 1.37·23-s + 4/5·25-s − 0.249·27-s + 0.185·29-s − 1.00·31-s − 0.996·33-s − 1.30·35-s + 0.533·37-s + 1.63·39-s − 1.19·41-s − 1.20·43-s + 1.09·45-s − 1.10·47-s − 0.0553·49-s − 0.845·51-s + 0.152·53-s − 0.992·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 29 | \( 1 - p T \) |
good | 3 | \( 1 - 7 T + p^{3} T^{2} \) |
| 5 | \( 1 - 3 p T + p^{3} T^{2} \) |
| 7 | \( 1 + 18 T + p^{3} T^{2} \) |
| 11 | \( 1 + 27 T + p^{3} T^{2} \) |
| 13 | \( 1 - 57 T + p^{3} T^{2} \) |
| 17 | \( 1 + 44 T + p^{3} T^{2} \) |
| 19 | \( 1 + 8 p T + p^{3} T^{2} \) |
| 23 | \( 1 + 152 T + p^{3} T^{2} \) |
| 31 | \( 1 + 173 T + p^{3} T^{2} \) |
| 37 | \( 1 - 120 T + p^{3} T^{2} \) |
| 41 | \( 1 + 314 T + p^{3} T^{2} \) |
| 43 | \( 1 + 339 T + p^{3} T^{2} \) |
| 47 | \( 1 + 357 T + p^{3} T^{2} \) |
| 53 | \( 1 - 59 T + p^{3} T^{2} \) |
| 59 | \( 1 - 572 T + p^{3} T^{2} \) |
| 61 | \( 1 - 420 T + p^{3} T^{2} \) |
| 67 | \( 1 + 660 T + p^{3} T^{2} \) |
| 71 | \( 1 - 726 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1004 T + p^{3} T^{2} \) |
| 79 | \( 1 - 361 T + p^{3} T^{2} \) |
| 83 | \( 1 - 168 T + p^{3} T^{2} \) |
| 89 | \( 1 - 58 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1206 T + p^{3} T^{2} \) |
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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
−8.507517666074057418935153720638, −8.083399329014209084791448663366, −6.66387743586518704712858192906, −6.30554282660061211594364324996, −5.37153569872933095148909918524, −4.03843168435977665445214540205, −3.30537393398408568837523910515, −2.26830459368609338507731532271, −1.85309194034289259384821634047, 0,
1.85309194034289259384821634047, 2.26830459368609338507731532271, 3.30537393398408568837523910515, 4.03843168435977665445214540205, 5.37153569872933095148909918524, 6.30554282660061211594364324996, 6.66387743586518704712858192906, 8.083399329014209084791448663366, 8.507517666074057418935153720638