L(s) = 1 | + 2.93·3-s + 5-s − 7-s + 5.60·9-s − 3.64·11-s − 1.06·13-s + 2.93·15-s − 7.78·17-s − 5.17·19-s − 2.93·21-s − 1.48·23-s + 25-s + 7.63·27-s − 2.54·29-s − 1.46·31-s − 10.7·33-s − 35-s − 4.03·37-s − 3.12·39-s − 3.98·41-s − 43-s + 5.60·45-s − 11.0·47-s + 49-s − 22.8·51-s + 5.93·53-s − 3.64·55-s + ⋯ |
L(s) = 1 | + 1.69·3-s + 0.447·5-s − 0.377·7-s + 1.86·9-s − 1.10·11-s − 0.295·13-s + 0.757·15-s − 1.88·17-s − 1.18·19-s − 0.640·21-s − 0.310·23-s + 0.200·25-s + 1.47·27-s − 0.472·29-s − 0.262·31-s − 1.86·33-s − 0.169·35-s − 0.662·37-s − 0.500·39-s − 0.622·41-s − 0.152·43-s + 0.835·45-s − 1.60·47-s + 0.142·49-s − 3.19·51-s + 0.814·53-s − 0.492·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 43 | \( 1 + T \) |
good | 3 | \( 1 - 2.93T + 3T^{2} \) |
| 11 | \( 1 + 3.64T + 11T^{2} \) |
| 13 | \( 1 + 1.06T + 13T^{2} \) |
| 17 | \( 1 + 7.78T + 17T^{2} \) |
| 19 | \( 1 + 5.17T + 19T^{2} \) |
| 23 | \( 1 + 1.48T + 23T^{2} \) |
| 29 | \( 1 + 2.54T + 29T^{2} \) |
| 31 | \( 1 + 1.46T + 31T^{2} \) |
| 37 | \( 1 + 4.03T + 37T^{2} \) |
| 41 | \( 1 + 3.98T + 41T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 - 5.93T + 53T^{2} \) |
| 59 | \( 1 - 9.51T + 59T^{2} \) |
| 61 | \( 1 - 3.35T + 61T^{2} \) |
| 67 | \( 1 - 7.17T + 67T^{2} \) |
| 71 | \( 1 - 0.889T + 71T^{2} \) |
| 73 | \( 1 + 6.96T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 - 8.62T + 89T^{2} \) |
| 97 | \( 1 + 7.03T + 97T^{2} \) |
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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
−7.981156863252141583658352488565, −6.98911224915819486552231254547, −6.65037451341783198249047521137, −5.51678654492982709758013701486, −4.62352585076263435188203985390, −3.92024661676683222887507165187, −3.05564087487660541826104710520, −2.26428737100412451199777271511, −1.94040744533698274526286351874, 0,
1.94040744533698274526286351874, 2.26428737100412451199777271511, 3.05564087487660541826104710520, 3.92024661676683222887507165187, 4.62352585076263435188203985390, 5.51678654492982709758013701486, 6.65037451341783198249047521137, 6.98911224915819486552231254547, 7.981156863252141583658352488565