| L(s) = 1 | − 2-s − 0.414·3-s + 4-s − 5-s + 0.414·6-s + 0.282·7-s − 8-s − 2.82·9-s + 10-s + 5.27·11-s − 0.414·12-s + 1.86·13-s − 0.282·14-s + 0.414·15-s + 16-s − 5.24·17-s + 2.82·18-s − 0.946·19-s − 20-s − 0.117·21-s − 5.27·22-s + 0.414·24-s + 25-s − 1.86·26-s + 2.41·27-s + 0.282·28-s − 1.64·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.239·3-s + 0.5·4-s − 0.447·5-s + 0.169·6-s + 0.106·7-s − 0.353·8-s − 0.942·9-s + 0.316·10-s + 1.59·11-s − 0.119·12-s + 0.516·13-s − 0.0755·14-s + 0.106·15-s + 0.250·16-s − 1.27·17-s + 0.666·18-s − 0.217·19-s − 0.223·20-s − 0.0255·21-s − 1.12·22-s + 0.0845·24-s + 0.200·25-s − 0.365·26-s + 0.464·27-s + 0.0533·28-s − 0.305·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 0.414T + 3T^{2} \) |
| 7 | \( 1 - 0.282T + 7T^{2} \) |
| 11 | \( 1 - 5.27T + 11T^{2} \) |
| 13 | \( 1 - 1.86T + 13T^{2} \) |
| 17 | \( 1 + 5.24T + 17T^{2} \) |
| 19 | \( 1 + 0.946T + 19T^{2} \) |
| 29 | \( 1 + 1.64T + 29T^{2} \) |
| 31 | \( 1 - 3.85T + 31T^{2} \) |
| 37 | \( 1 - 0.828T + 37T^{2} \) |
| 41 | \( 1 + 7.53T + 41T^{2} \) |
| 43 | \( 1 + 0.663T + 43T^{2} \) |
| 47 | \( 1 - 0.353T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 - 3T + 59T^{2} \) |
| 61 | \( 1 - 9.75T + 61T^{2} \) |
| 67 | \( 1 - 15.0T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 + 15.9T + 73T^{2} \) |
| 79 | \( 1 + 4.81T + 79T^{2} \) |
| 83 | \( 1 - 0.643T + 83T^{2} \) |
| 89 | \( 1 + 2.29T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−8.257443536358055832939088121939, −6.90074421054162436972864972789, −6.63369077357995045026941581919, −5.91302545484337308834837495394, −4.89983964209282161086583761769, −4.02261723626437147378572980877, −3.28021069616111278212749268523, −2.19788786259221328474722412606, −1.18183971573900564333980427442, 0,
1.18183971573900564333980427442, 2.19788786259221328474722412606, 3.28021069616111278212749268523, 4.02261723626437147378572980877, 4.89983964209282161086583761769, 5.91302545484337308834837495394, 6.63369077357995045026941581919, 6.90074421054162436972864972789, 8.257443536358055832939088121939
