| L(s) = 1 | − 2-s + 1.78·3-s + 4-s + 2.00·5-s − 1.78·6-s − 0.442·7-s − 8-s + 0.172·9-s − 2.00·10-s − 0.197·11-s + 1.78·12-s − 3.50·13-s + 0.442·14-s + 3.57·15-s + 16-s − 3.35·17-s − 0.172·18-s − 1.01·19-s + 2.00·20-s − 0.787·21-s + 0.197·22-s + 1.79·23-s − 1.78·24-s − 0.978·25-s + 3.50·26-s − 5.03·27-s − 0.442·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.02·3-s + 0.5·4-s + 0.896·5-s − 0.727·6-s − 0.167·7-s − 0.353·8-s + 0.0576·9-s − 0.634·10-s − 0.0595·11-s + 0.514·12-s − 0.972·13-s + 0.118·14-s + 0.922·15-s + 0.250·16-s − 0.813·17-s − 0.0407·18-s − 0.232·19-s + 0.448·20-s − 0.171·21-s + 0.0420·22-s + 0.374·23-s − 0.363·24-s − 0.195·25-s + 0.687·26-s − 0.969·27-s − 0.0835·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 2017 | \( 1 + T \) |
| good | 3 | \( 1 - 1.78T + 3T^{2} \) |
| 5 | \( 1 - 2.00T + 5T^{2} \) |
| 7 | \( 1 + 0.442T + 7T^{2} \) |
| 11 | \( 1 + 0.197T + 11T^{2} \) |
| 13 | \( 1 + 3.50T + 13T^{2} \) |
| 17 | \( 1 + 3.35T + 17T^{2} \) |
| 19 | \( 1 + 1.01T + 19T^{2} \) |
| 23 | \( 1 - 1.79T + 23T^{2} \) |
| 29 | \( 1 + 7.05T + 29T^{2} \) |
| 31 | \( 1 + 1.31T + 31T^{2} \) |
| 37 | \( 1 + 5.16T + 37T^{2} \) |
| 41 | \( 1 - 2.93T + 41T^{2} \) |
| 43 | \( 1 - 3.97T + 43T^{2} \) |
| 47 | \( 1 - 6.15T + 47T^{2} \) |
| 53 | \( 1 + 9.08T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 - 8.09T + 61T^{2} \) |
| 67 | \( 1 - 5.29T + 67T^{2} \) |
| 71 | \( 1 - 1.67T + 71T^{2} \) |
| 73 | \( 1 - 3.03T + 73T^{2} \) |
| 79 | \( 1 - 1.47T + 79T^{2} \) |
| 83 | \( 1 + 4.14T + 83T^{2} \) |
| 89 | \( 1 + 8.54T + 89T^{2} \) |
| 97 | \( 1 + 16.0T + 97T^{2} \) |
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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
−8.193689080156170894380702661895, −7.49349377703406960189806883901, −6.82650872415366848751825422892, −5.94855580746736674196768208657, −5.22540552912843230508087549932, −4.08139626509249756513973784240, −3.06082695312827851601742451767, −2.32481191198554881793996090905, −1.73503128510396390743346203457, 0,
1.73503128510396390743346203457, 2.32481191198554881793996090905, 3.06082695312827851601742451767, 4.08139626509249756513973784240, 5.22540552912843230508087549932, 5.94855580746736674196768208657, 6.82650872415366848751825422892, 7.49349377703406960189806883901, 8.193689080156170894380702661895
