| L(s) = 1 | − 2.61·2-s − 0.0204·3-s − 1.17·4-s − 12.1·5-s + 0.0534·6-s + 23.9·8-s − 26.9·9-s + 31.8·10-s − 39.3·11-s + 0.0240·12-s + 5.08·13-s + 0.249·15-s − 53.2·16-s + 71.6·17-s + 70.5·18-s − 52.9·19-s + 14.3·20-s + 102.·22-s − 16.9·23-s − 0.490·24-s + 23.7·25-s − 13.2·26-s + 1.10·27-s − 138.·29-s − 0.652·30-s + 208.·31-s − 52.6·32-s + ⋯ |
| L(s) = 1 | − 0.923·2-s − 0.00393·3-s − 0.146·4-s − 1.09·5-s + 0.00363·6-s + 1.05·8-s − 0.999·9-s + 1.00·10-s − 1.07·11-s + 0.000578·12-s + 0.108·13-s + 0.00429·15-s − 0.831·16-s + 1.02·17-s + 0.923·18-s − 0.639·19-s + 0.160·20-s + 0.996·22-s − 0.153·23-s − 0.00417·24-s + 0.190·25-s − 0.100·26-s + 0.00787·27-s − 0.888·29-s − 0.00396·30-s + 1.20·31-s − 0.291·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 41 | \( 1 + 41T \) |
| good | 2 | \( 1 + 2.61T + 8T^{2} \) |
| 3 | \( 1 + 0.0204T + 27T^{2} \) |
| 5 | \( 1 + 12.1T + 125T^{2} \) |
| 11 | \( 1 + 39.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 5.08T + 2.19e3T^{2} \) |
| 17 | \( 1 - 71.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 52.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 16.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 138.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 208.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 242.T + 5.06e4T^{2} \) |
| 43 | \( 1 - 395.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 483.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 287.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 149.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 499.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 253.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 219.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 764.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.31e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 266.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.63e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.30e3T + 9.12e5T^{2} \) |
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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
−8.474041471684455810751751098239, −7.70640709874190331989414482985, −7.40340987526384391944252335645, −6.00261226471677917489516808474, −5.19108171970867086854567780813, −4.25042832247084032362004782792, −3.37075808940091374588933253637, −2.26983013262180230223153413159, −0.77857406434082352416025192094, 0,
0.77857406434082352416025192094, 2.26983013262180230223153413159, 3.37075808940091374588933253637, 4.25042832247084032362004782792, 5.19108171970867086854567780813, 6.00261226471677917489516808474, 7.40340987526384391944252335645, 7.70640709874190331989414482985, 8.474041471684455810751751098239
