| L(s) = 1 | + 2.51e10·2-s − 1.85e20·3-s − 3.86e25·4-s + 5.79e28·5-s − 4.64e30·6-s − 1.94e31·7-s − 1.94e36·8-s − 1.62e39·9-s + 1.45e39·10-s + 7.74e43·11-s + 7.16e45·12-s + 1.75e47·13-s − 4.87e41·14-s − 1.07e49·15-s + 1.49e51·16-s + 2.92e52·17-s − 4.07e49·18-s − 2.00e54·19-s − 2.24e54·20-s + 3.59e51·21-s + 1.94e54·22-s + 5.01e57·23-s + 3.59e56·24-s − 2.55e59·25-s + 4.41e57·26-s + 6.95e60·27-s + 7.51e56·28-s + ⋯ |
| L(s) = 1 | + 0.00403·2-s − 0.977·3-s − 0.999·4-s + 0.114·5-s − 0.00394·6-s − 2.35e − 5·7-s − 0.00807·8-s − 0.0451·9-s + 0.000460·10-s + 0.426·11-s + 0.977·12-s + 0.799·13-s − 0.111·15-s + 0.999·16-s + 1.48·17-s − 0.000182·18-s − 0.900·19-s − 0.114·20-s + 2.29e−5·21-s + 0.00172·22-s + 0.670·23-s + 0.00788·24-s − 0.986·25-s + 0.00322·26-s + 1.02·27-s + 2.35e−5·28-s − 0.913·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(86-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+85/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(43)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{87}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 2.51e10T + 3.86e25T^{2} \) |
| 3 | \( 1 + 1.85e20T + 3.59e40T^{2} \) |
| 5 | \( 1 - 5.79e28T + 2.58e59T^{2} \) |
| 7 | \( 1 + 1.94e31T + 6.81e71T^{2} \) |
| 11 | \( 1 - 7.74e43T + 3.29e88T^{2} \) |
| 13 | \( 1 - 1.75e47T + 4.84e94T^{2} \) |
| 17 | \( 1 - 2.92e52T + 3.87e104T^{2} \) |
| 19 | \( 1 + 2.00e54T + 4.94e108T^{2} \) |
| 23 | \( 1 - 5.01e57T + 5.58e115T^{2} \) |
| 29 | \( 1 + 1.29e62T + 2.01e124T^{2} \) |
| 31 | \( 1 - 1.68e63T + 5.83e126T^{2} \) |
| 37 | \( 1 - 1.08e66T + 1.98e133T^{2} \) |
| 41 | \( 1 - 1.78e68T + 1.22e137T^{2} \) |
| 43 | \( 1 + 3.64e69T + 6.99e138T^{2} \) |
| 47 | \( 1 + 2.03e71T + 1.34e142T^{2} \) |
| 53 | \( 1 + 1.76e73T + 3.65e146T^{2} \) |
| 59 | \( 1 - 2.01e75T + 3.32e150T^{2} \) |
| 61 | \( 1 - 1.24e76T + 5.66e151T^{2} \) |
| 67 | \( 1 + 1.97e77T + 1.64e155T^{2} \) |
| 71 | \( 1 - 5.54e78T + 2.27e157T^{2} \) |
| 73 | \( 1 - 3.96e78T + 2.41e158T^{2} \) |
| 79 | \( 1 + 5.29e80T + 1.98e161T^{2} \) |
| 83 | \( 1 + 2.15e81T + 1.32e163T^{2} \) |
| 89 | \( 1 - 1.00e83T + 4.99e165T^{2} \) |
| 97 | \( 1 + 3.56e84T + 7.50e168T^{2} \) |
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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
−14.49934608555727710379490473208, −12.95525385837688860086675247808, −11.46324743893237439043814033865, −9.878233763647951484299689491738, −8.306314550522273966186358870031, −6.19360691239888574386956191290, −5.08627527979381804968844224839, −3.59182721586255954855468217714, −1.20479656830402345073707867603, 0,
1.20479656830402345073707867603, 3.59182721586255954855468217714, 5.08627527979381804968844224839, 6.19360691239888574386956191290, 8.306314550522273966186358870031, 9.878233763647951484299689491738, 11.46324743893237439043814033865, 12.95525385837688860086675247808, 14.49934608555727710379490473208
