| L(s) = 1 | − 2.17·2-s + 1.70·3-s + 2.70·4-s − 3.70·6-s − 1.07·7-s − 1.53·8-s − 0.0783·9-s − 6.34·11-s + 4.63·12-s − 1.36·13-s + 2.34·14-s − 2.07·16-s − 3.26·17-s + 0.170·18-s − 19-s − 1.84·21-s + 13.7·22-s − 2.34·23-s − 2.63·24-s + 2.97·26-s − 5.26·27-s − 2.92·28-s + 1.41·29-s + 8.68·31-s + 7.58·32-s − 10.8·33-s + 7.07·34-s + ⋯ |
| L(s) = 1 | − 1.53·2-s + 0.986·3-s + 1.35·4-s − 1.51·6-s − 0.407·7-s − 0.544·8-s − 0.0261·9-s − 1.91·11-s + 1.33·12-s − 0.379·13-s + 0.625·14-s − 0.519·16-s − 0.791·17-s + 0.0400·18-s − 0.229·19-s − 0.402·21-s + 2.93·22-s − 0.487·23-s − 0.537·24-s + 0.582·26-s − 1.01·27-s − 0.552·28-s + 0.263·29-s + 1.55·31-s + 1.34·32-s − 1.88·33-s + 1.21·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| good | 2 | \( 1 + 2.17T + 2T^{2} \) |
| 3 | \( 1 - 1.70T + 3T^{2} \) |
| 7 | \( 1 + 1.07T + 7T^{2} \) |
| 11 | \( 1 + 6.34T + 11T^{2} \) |
| 13 | \( 1 + 1.36T + 13T^{2} \) |
| 17 | \( 1 + 3.26T + 17T^{2} \) |
| 23 | \( 1 + 2.34T + 23T^{2} \) |
| 29 | \( 1 - 1.41T + 29T^{2} \) |
| 31 | \( 1 - 8.68T + 31T^{2} \) |
| 37 | \( 1 + 5.36T + 37T^{2} \) |
| 41 | \( 1 + 3.26T + 41T^{2} \) |
| 43 | \( 1 - 11.9T + 43T^{2} \) |
| 47 | \( 1 + 1.07T + 47T^{2} \) |
| 53 | \( 1 + 6.63T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 - 5.60T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 + 5.41T + 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 - 7.57T + 89T^{2} \) |
| 97 | \( 1 - 8.88T + 97T^{2} \) |
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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
−10.31803516712566130100771050541, −9.538426248023707266528190880262, −8.726054419556630194650702462318, −8.028091385468656552884780165313, −7.48024771249110438457370552182, −6.24145341271270623921407924606, −4.73287086874634028716202162561, −2.96067152540332391416046639408, −2.18381579025341456127530937661, 0,
2.18381579025341456127530937661, 2.96067152540332391416046639408, 4.73287086874634028716202162561, 6.24145341271270623921407924606, 7.48024771249110438457370552182, 8.028091385468656552884780165313, 8.726054419556630194650702462318, 9.538426248023707266528190880262, 10.31803516712566130100771050541
