| L(s) = 1 | + 2·2-s + 0.405·3-s + 4·4-s − 6.36·5-s + 0.811·6-s + 2.55·7-s + 8·8-s − 26.8·9-s − 12.7·10-s − 26.1·11-s + 1.62·12-s + 5.10·14-s − 2.58·15-s + 16·16-s − 93.7·17-s − 53.6·18-s + 37.2·19-s − 25.4·20-s + 1.03·21-s − 52.2·22-s − 104.·23-s + 3.24·24-s − 84.5·25-s − 21.8·27-s + 10.2·28-s + 249.·29-s − 5.16·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.0780·3-s + 0.5·4-s − 0.569·5-s + 0.0552·6-s + 0.137·7-s + 0.353·8-s − 0.993·9-s − 0.402·10-s − 0.715·11-s + 0.0390·12-s + 0.0973·14-s − 0.0444·15-s + 0.250·16-s − 1.33·17-s − 0.702·18-s + 0.449·19-s − 0.284·20-s + 0.0107·21-s − 0.506·22-s − 0.951·23-s + 0.0276·24-s − 0.676·25-s − 0.155·27-s + 0.0688·28-s + 1.59·29-s − 0.0314·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 0.405T + 27T^{2} \) |
| 5 | \( 1 + 6.36T + 125T^{2} \) |
| 7 | \( 1 - 2.55T + 343T^{2} \) |
| 11 | \( 1 + 26.1T + 1.33e3T^{2} \) |
| 17 | \( 1 + 93.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 37.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 104.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 249.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 278.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 10.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 371.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 413.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 238.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 424.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 774.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 123.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 881.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 118.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 209.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 532.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 376.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 42.6T + 7.04e5T^{2} \) |
| 97 | \( 1 + 639.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.05900312009442295955224518681, −9.835375926414247106941075482605, −8.529225443376204994208338109994, −7.83267649050432889229677083006, −6.64644971811449574893379555639, −5.59057682703643727938770927596, −4.56836720316319430840028186609, −3.39292206670555417064824648698, −2.21994418057166274676689482118, 0,
2.21994418057166274676689482118, 3.39292206670555417064824648698, 4.56836720316319430840028186609, 5.59057682703643727938770927596, 6.64644971811449574893379555639, 7.83267649050432889229677083006, 8.529225443376204994208338109994, 9.835375926414247106941075482605, 11.05900312009442295955224518681
