| L(s) = 1 | − 16·2-s − 7.07·3-s + 256·4-s − 302.·5-s + 113.·6-s − 6.51e3·7-s − 4.09e3·8-s − 1.96e4·9-s + 4.83e3·10-s + 2.26e3·11-s − 1.81e3·12-s + 1.04e5·14-s + 2.13e3·15-s + 6.55e4·16-s + 2.41e5·17-s + 3.14e5·18-s + 3.25e5·19-s − 7.73e4·20-s + 4.61e4·21-s − 3.62e4·22-s − 1.66e6·23-s + 2.89e4·24-s − 1.86e6·25-s + 2.78e5·27-s − 1.66e6·28-s − 2.69e6·29-s − 3.42e4·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.0504·3-s + 0.5·4-s − 0.216·5-s + 0.0356·6-s − 1.02·7-s − 0.353·8-s − 0.997·9-s + 0.152·10-s + 0.0466·11-s − 0.0252·12-s + 0.725·14-s + 0.0109·15-s + 0.250·16-s + 0.700·17-s + 0.705·18-s + 0.572·19-s − 0.108·20-s + 0.0517·21-s − 0.0330·22-s − 1.24·23-s + 0.0178·24-s − 0.953·25-s + 0.100·27-s − 0.513·28-s − 0.708·29-s − 0.00771·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.2797440001\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2797440001\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 16T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 7.07T + 1.96e4T^{2} \) |
| 5 | \( 1 + 302.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 6.51e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 2.26e3T + 2.35e9T^{2} \) |
| 17 | \( 1 - 2.41e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 3.25e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.66e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.69e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 3.21e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.87e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.25e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.74e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 9.41e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.21e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 3.34e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.94e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 6.02e6T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.31e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.33e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.86e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.19e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + 3.13e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 5.74e8T + 7.60e17T^{2} \) |
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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
−9.850314375474342965532917933430, −9.180131680804505919383022619774, −8.159409134427570318745511683661, −7.33564853519223837266653283275, −6.19661777482389420830655710244, −5.49859364023416637678977728095, −3.75559930339246849296605929995, −2.95864733467541088608850064409, −1.70177194135466718807702295700, −0.24536968273860115236784909218,
0.24536968273860115236784909218, 1.70177194135466718807702295700, 2.95864733467541088608850064409, 3.75559930339246849296605929995, 5.49859364023416637678977728095, 6.19661777482389420830655710244, 7.33564853519223837266653283275, 8.159409134427570318745511683661, 9.180131680804505919383022619774, 9.850314375474342965532917933430
