| L(s) = 1 | − 1.61·2-s + 0.614·4-s − 1.46·5-s + 3.83·7-s + 2.24·8-s + 2.37·10-s − 2.90·11-s + 2.33·13-s − 6.20·14-s − 4.85·16-s − 3.77·17-s − 1.21·19-s − 0.901·20-s + 4.69·22-s + 1.47·23-s − 2.85·25-s − 3.76·26-s + 2.35·28-s − 1.02·29-s − 1.72·31-s + 3.36·32-s + 6.09·34-s − 5.62·35-s − 1.08·37-s + 1.96·38-s − 3.28·40-s + 1.58·41-s + ⋯ |
| L(s) = 1 | − 1.14·2-s + 0.307·4-s − 0.655·5-s + 1.44·7-s + 0.791·8-s + 0.749·10-s − 0.875·11-s + 0.646·13-s − 1.65·14-s − 1.21·16-s − 0.914·17-s − 0.279·19-s − 0.201·20-s + 1.00·22-s + 0.307·23-s − 0.570·25-s − 0.739·26-s + 0.445·28-s − 0.190·29-s − 0.308·31-s + 0.594·32-s + 1.04·34-s − 0.950·35-s − 0.179·37-s + 0.319·38-s − 0.519·40-s + 0.246·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 239 | \( 1 + T \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 5 | \( 1 + 1.46T + 5T^{2} \) |
| 7 | \( 1 - 3.83T + 7T^{2} \) |
| 11 | \( 1 + 2.90T + 11T^{2} \) |
| 13 | \( 1 - 2.33T + 13T^{2} \) |
| 17 | \( 1 + 3.77T + 17T^{2} \) |
| 19 | \( 1 + 1.21T + 19T^{2} \) |
| 23 | \( 1 - 1.47T + 23T^{2} \) |
| 29 | \( 1 + 1.02T + 29T^{2} \) |
| 31 | \( 1 + 1.72T + 31T^{2} \) |
| 37 | \( 1 + 1.08T + 37T^{2} \) |
| 41 | \( 1 - 1.58T + 41T^{2} \) |
| 43 | \( 1 - 0.433T + 43T^{2} \) |
| 47 | \( 1 + 7.65T + 47T^{2} \) |
| 53 | \( 1 + 1.07T + 53T^{2} \) |
| 59 | \( 1 - 4.82T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 + 6.11T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 - 6.08T + 89T^{2} \) |
| 97 | \( 1 - 7.99T + 97T^{2} \) |
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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
−8.587102340578765836069106283239, −8.000242154919240874909105215855, −7.61549174006609877732476472577, −6.66129794722342781777067522628, −5.37179270836595936558916040011, −4.64360366605365318322439112610, −3.87061547662322498004781917566, −2.32645096457765236043841735274, −1.37440404473971757334133741069, 0,
1.37440404473971757334133741069, 2.32645096457765236043841735274, 3.87061547662322498004781917566, 4.64360366605365318322439112610, 5.37179270836595936558916040011, 6.66129794722342781777067522628, 7.61549174006609877732476472577, 8.000242154919240874909105215855, 8.587102340578765836069106283239
