| L(s) = 1 | − 6.04e12·2-s + 9.44e19·3-s + 2.69e25·4-s − 3.42e28·5-s − 5.71e32·6-s − 1.33e34·7-s − 1.04e38·8-s + 4.92e39·9-s + 2.07e41·10-s + 1.77e43·11-s + 2.54e45·12-s − 1.23e46·13-s + 8.10e46·14-s − 3.23e48·15-s + 3.70e50·16-s + 3.64e50·17-s − 2.98e52·18-s + 4.63e52·19-s − 9.21e53·20-s − 1.26e54·21-s − 1.07e56·22-s + 2.40e56·23-s − 9.84e57·24-s − 9.16e57·25-s + 7.48e58·26-s + 8.86e58·27-s − 3.60e59·28-s + ⋯ |
| L(s) = 1 | − 1.94·2-s + 1.49·3-s + 2.78·4-s − 0.336·5-s − 2.90·6-s − 0.113·7-s − 3.46·8-s + 1.23·9-s + 0.655·10-s + 1.07·11-s + 4.15·12-s − 0.731·13-s + 0.220·14-s − 0.503·15-s + 3.95·16-s + 0.314·17-s − 2.40·18-s + 0.396·19-s − 0.937·20-s − 0.169·21-s − 2.09·22-s + 0.741·23-s − 5.18·24-s − 0.886·25-s + 1.42·26-s + 0.351·27-s − 0.316·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(84-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+83/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(42)\) |
\(\approx\) |
\(1.441479264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.441479264\) |
| \(L(\frac{85}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 6.04e12T + 9.67e24T^{2} \) |
| 3 | \( 1 - 9.44e19T + 3.99e39T^{2} \) |
| 5 | \( 1 + 3.42e28T + 1.03e58T^{2} \) |
| 7 | \( 1 + 1.33e34T + 1.39e70T^{2} \) |
| 11 | \( 1 - 1.77e43T + 2.72e86T^{2} \) |
| 13 | \( 1 + 1.23e46T + 2.86e92T^{2} \) |
| 17 | \( 1 - 3.64e50T + 1.34e102T^{2} \) |
| 19 | \( 1 - 4.63e52T + 1.36e106T^{2} \) |
| 23 | \( 1 - 2.40e56T + 1.05e113T^{2} \) |
| 29 | \( 1 - 8.96e60T + 2.39e121T^{2} \) |
| 31 | \( 1 - 2.68e61T + 6.06e123T^{2} \) |
| 37 | \( 1 + 1.51e65T + 1.44e130T^{2} \) |
| 41 | \( 1 - 2.47e66T + 7.26e133T^{2} \) |
| 43 | \( 1 - 5.12e67T + 3.78e135T^{2} \) |
| 47 | \( 1 + 1.75e69T + 6.08e138T^{2} \) |
| 53 | \( 1 - 3.40e71T + 1.30e143T^{2} \) |
| 59 | \( 1 - 1.58e73T + 9.56e146T^{2} \) |
| 61 | \( 1 - 2.20e74T + 1.52e148T^{2} \) |
| 67 | \( 1 + 2.70e74T + 3.66e151T^{2} \) |
| 71 | \( 1 - 5.27e76T + 4.51e153T^{2} \) |
| 73 | \( 1 + 1.81e77T + 4.52e154T^{2} \) |
| 79 | \( 1 - 5.15e78T + 3.18e157T^{2} \) |
| 83 | \( 1 - 4.95e79T + 1.92e159T^{2} \) |
| 89 | \( 1 - 5.60e80T + 6.30e161T^{2} \) |
| 97 | \( 1 - 3.88e82T + 7.98e164T^{2} \) |
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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
−15.85234291751269487652239409512, −14.54823911282830429837245361694, −11.87732052149452396577936824670, −9.982455467055416185137742120150, −9.000709736037713581275729398077, −7.993457430174405812237471046184, −6.85859965786130239287972814612, −3.32389226197368291992344336393, −2.15851733495935603386191137335, −0.878639372700450754888891206178,
0.878639372700450754888891206178, 2.15851733495935603386191137335, 3.32389226197368291992344336393, 6.85859965786130239287972814612, 7.993457430174405812237471046184, 9.000709736037713581275729398077, 9.982455467055416185137742120150, 11.87732052149452396577936824670, 14.54823911282830429837245361694, 15.85234291751269487652239409512
