| L(s) = 1 | − 4.31·2-s + 10.6·4-s + 5·5-s − 11.3·8-s − 21.5·10-s − 19.7·11-s − 71.3·13-s − 36.0·16-s − 31.3·17-s − 136.·19-s + 53.1·20-s + 85.1·22-s + 100.·23-s + 25·25-s + 307.·26-s + 288.·29-s + 208.·31-s + 246.·32-s + 135.·34-s + 309.·37-s + 588.·38-s − 56.8·40-s − 181.·41-s − 18.2·43-s − 209.·44-s − 435.·46-s − 147.·47-s + ⋯ |
| L(s) = 1 | − 1.52·2-s + 1.32·4-s + 0.447·5-s − 0.502·8-s − 0.682·10-s − 0.540·11-s − 1.52·13-s − 0.562·16-s − 0.447·17-s − 1.64·19-s + 0.594·20-s + 0.825·22-s + 0.914·23-s + 0.200·25-s + 2.32·26-s + 1.84·29-s + 1.21·31-s + 1.36·32-s + 0.682·34-s + 1.37·37-s + 2.51·38-s − 0.224·40-s − 0.691·41-s − 0.0645·43-s − 0.718·44-s − 1.39·46-s − 0.458·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 4.31T + 8T^{2} \) |
| 11 | \( 1 + 19.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 71.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 31.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 136.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 100.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 288.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 208.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 309.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 181.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 18.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 147.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 127.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 322.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 341.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 84.3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 315.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.09e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.23e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 643.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.14e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.41e3T + 9.12e5T^{2} \) |
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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
−8.308804107412073013538911672436, −7.88079029635510478006113688897, −6.77087147665941103360823357633, −6.49496611055088818601213585915, −5.05077000555621080716121651416, −4.46054636719244234525752980371, −2.67237947422684543147802421394, −2.26287490741642889037184638622, −0.970572541258496680490855146850, 0,
0.970572541258496680490855146850, 2.26287490741642889037184638622, 2.67237947422684543147802421394, 4.46054636719244234525752980371, 5.05077000555621080716121651416, 6.49496611055088818601213585915, 6.77087147665941103360823357633, 7.88079029635510478006113688897, 8.308804107412073013538911672436
