L(s) = 1 | − 3.33·3-s + 5-s + 2.35·7-s + 8.13·9-s − 5.41·11-s + 7.10·13-s − 3.33·15-s − 2.30·17-s + 3.33·19-s − 7.85·21-s − 2.59·23-s + 25-s − 17.1·27-s − 1.45·29-s + 2.14·31-s + 18.0·33-s + 2.35·35-s − 8.97·37-s − 23.7·39-s + 0.901·41-s − 1.97·43-s + 8.13·45-s − 11.7·47-s − 1.46·49-s + 7.69·51-s + 10.0·53-s − 5.41·55-s + ⋯ |
L(s) = 1 | − 1.92·3-s + 0.447·5-s + 0.889·7-s + 2.71·9-s − 1.63·11-s + 1.97·13-s − 0.861·15-s − 0.559·17-s + 0.765·19-s − 1.71·21-s − 0.541·23-s + 0.200·25-s − 3.29·27-s − 0.271·29-s + 0.385·31-s + 3.14·33-s + 0.397·35-s − 1.47·37-s − 3.79·39-s + 0.140·41-s − 0.301·43-s + 1.21·45-s − 1.71·47-s − 0.209·49-s + 1.07·51-s + 1.37·53-s − 0.729·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 - T \) |
good | 3 | \( 1 + 3.33T + 3T^{2} \) |
| 7 | \( 1 - 2.35T + 7T^{2} \) |
| 11 | \( 1 + 5.41T + 11T^{2} \) |
| 13 | \( 1 - 7.10T + 13T^{2} \) |
| 17 | \( 1 + 2.30T + 17T^{2} \) |
| 19 | \( 1 - 3.33T + 19T^{2} \) |
| 23 | \( 1 + 2.59T + 23T^{2} \) |
| 29 | \( 1 + 1.45T + 29T^{2} \) |
| 31 | \( 1 - 2.14T + 31T^{2} \) |
| 37 | \( 1 + 8.97T + 37T^{2} \) |
| 41 | \( 1 - 0.901T + 41T^{2} \) |
| 43 | \( 1 + 1.97T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 + 4.21T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 - 0.385T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 + 6.72T + 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 - 2.69T + 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 - 5.95T + 97T^{2} \) |
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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
−7.66065631983843820231351750607, −6.77580002105806219226991707983, −6.12814002794764713683714138074, −5.55928411265913408147291036363, −5.05640380484517641352097661317, −4.42547314409810870690633108874, −3.37255595062065185746513183012, −1.89399004432108335901745227681, −1.21891355678433128626333278294, 0,
1.21891355678433128626333278294, 1.89399004432108335901745227681, 3.37255595062065185746513183012, 4.42547314409810870690633108874, 5.05640380484517641352097661317, 5.55928411265913408147291036363, 6.12814002794764713683714138074, 6.77580002105806219226991707983, 7.66065631983843820231351750607