L(s) = 1 | + 1.93·2-s + 2.41·3-s + 1.74·4-s + 4.67·6-s − 1.46·7-s − 0.487·8-s + 2.82·9-s − 2.89·11-s + 4.22·12-s − 6.12·13-s − 2.82·14-s − 4.44·16-s + 6.29·17-s + 5.47·18-s − 3.52·21-s − 5.60·22-s − 0.508·23-s − 1.17·24-s − 11.8·26-s − 0.414·27-s − 2.55·28-s − 1.72·29-s + 8.44·31-s − 7.62·32-s − 6.98·33-s + 12.1·34-s + 4.94·36-s + ⋯ |
L(s) = 1 | + 1.36·2-s + 1.39·3-s + 0.874·4-s + 1.90·6-s − 0.552·7-s − 0.172·8-s + 0.942·9-s − 0.872·11-s + 1.21·12-s − 1.69·13-s − 0.755·14-s − 1.11·16-s + 1.52·17-s + 1.29·18-s − 0.769·21-s − 1.19·22-s − 0.106·23-s − 0.240·24-s − 2.32·26-s − 0.0796·27-s − 0.482·28-s − 0.319·29-s + 1.51·31-s − 1.34·32-s − 1.21·33-s + 2.09·34-s + 0.824·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 - 1.93T + 2T^{2} \) |
| 3 | \( 1 - 2.41T + 3T^{2} \) |
| 7 | \( 1 + 1.46T + 7T^{2} \) |
| 11 | \( 1 + 2.89T + 11T^{2} \) |
| 13 | \( 1 + 6.12T + 13T^{2} \) |
| 17 | \( 1 - 6.29T + 17T^{2} \) |
| 23 | \( 1 + 0.508T + 23T^{2} \) |
| 29 | \( 1 + 1.72T + 29T^{2} \) |
| 31 | \( 1 - 8.44T + 31T^{2} \) |
| 37 | \( 1 + 3.13T + 37T^{2} \) |
| 41 | \( 1 + 7.44T + 41T^{2} \) |
| 43 | \( 1 + 6.90T + 43T^{2} \) |
| 47 | \( 1 - 0.316T + 47T^{2} \) |
| 53 | \( 1 + 4.34T + 53T^{2} \) |
| 59 | \( 1 + 2.25T + 59T^{2} \) |
| 61 | \( 1 + 6.29T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 - 6.63T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 + 14.6T + 79T^{2} \) |
| 83 | \( 1 - 4.94T + 83T^{2} \) |
| 89 | \( 1 - 5.35T + 89T^{2} \) |
| 97 | \( 1 + 15.9T + 97T^{2} \) |
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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
−7.35509697626666547524047249362, −6.73361712857419417247324742779, −5.80804773143043311113869079412, −5.12625198372876926883240221318, −4.60892831304722027288319061733, −3.65402267301500184244680482564, −3.01452628292470403391319061038, −2.76772996080782625982506385476, −1.82672528952274915031887333538, 0,
1.82672528952274915031887333538, 2.76772996080782625982506385476, 3.01452628292470403391319061038, 3.65402267301500184244680482564, 4.60892831304722027288319061733, 5.12625198372876926883240221318, 5.80804773143043311113869079412, 6.73361712857419417247324742779, 7.35509697626666547524047249362