| L(s) = 1 | − 2.43·2-s − 1.33·3-s + 3.94·4-s − 4.10·5-s + 3.25·6-s + 4.95·7-s − 4.74·8-s − 1.22·9-s + 10.0·10-s − 1.82·11-s − 5.26·12-s − 3.65·13-s − 12.0·14-s + 5.46·15-s + 3.68·16-s + 5.58·17-s + 2.98·18-s + 2.06·19-s − 16.1·20-s − 6.61·21-s + 4.44·22-s − 1.26·23-s + 6.32·24-s + 11.8·25-s + 8.91·26-s + 5.62·27-s + 19.5·28-s + ⋯ |
| L(s) = 1 | − 1.72·2-s − 0.769·3-s + 1.97·4-s − 1.83·5-s + 1.32·6-s + 1.87·7-s − 1.67·8-s − 0.407·9-s + 3.16·10-s − 0.549·11-s − 1.51·12-s − 1.01·13-s − 3.23·14-s + 1.41·15-s + 0.920·16-s + 1.35·17-s + 0.702·18-s + 0.474·19-s − 3.61·20-s − 1.44·21-s + 0.947·22-s − 0.263·23-s + 1.29·24-s + 2.36·25-s + 1.74·26-s + 1.08·27-s + 3.69·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 787 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 787 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 787 | \( 1 + T \) |
| good | 2 | \( 1 + 2.43T + 2T^{2} \) |
| 3 | \( 1 + 1.33T + 3T^{2} \) |
| 5 | \( 1 + 4.10T + 5T^{2} \) |
| 7 | \( 1 - 4.95T + 7T^{2} \) |
| 11 | \( 1 + 1.82T + 11T^{2} \) |
| 13 | \( 1 + 3.65T + 13T^{2} \) |
| 17 | \( 1 - 5.58T + 17T^{2} \) |
| 19 | \( 1 - 2.06T + 19T^{2} \) |
| 23 | \( 1 + 1.26T + 23T^{2} \) |
| 29 | \( 1 - 8.92T + 29T^{2} \) |
| 31 | \( 1 + 1.70T + 31T^{2} \) |
| 37 | \( 1 + 9.07T + 37T^{2} \) |
| 41 | \( 1 - 3.78T + 41T^{2} \) |
| 43 | \( 1 + 3.83T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 + 5.66T + 53T^{2} \) |
| 59 | \( 1 - 4.27T + 59T^{2} \) |
| 61 | \( 1 + 1.30T + 61T^{2} \) |
| 67 | \( 1 - 6.23T + 67T^{2} \) |
| 71 | \( 1 + 3.67T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 + 7.89T + 79T^{2} \) |
| 83 | \( 1 + 17.7T + 83T^{2} \) |
| 89 | \( 1 + 2.07T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−10.06898767428682792276956800727, −8.595600278317242591041403856100, −8.172112682263549297196248917304, −7.64691231770859741645291601401, −6.97423773406575787742549212121, −5.33092059356258587598872439288, −4.61536105479709525092125518824, −2.93134301225148915324998180102, −1.24951360189730967750487546951, 0,
1.24951360189730967750487546951, 2.93134301225148915324998180102, 4.61536105479709525092125518824, 5.33092059356258587598872439288, 6.97423773406575787742549212121, 7.64691231770859741645291601401, 8.172112682263549297196248917304, 8.595600278317242591041403856100, 10.06898767428682792276956800727
