L(s) = 1 | + 1.32·2-s + 0.433·3-s − 0.244·4-s − 1.11·5-s + 0.574·6-s − 1.01·7-s − 2.97·8-s − 2.81·9-s − 1.47·10-s − 3.85·11-s − 0.106·12-s − 5.34·13-s − 1.34·14-s − 0.482·15-s − 3.45·16-s + 6.52·17-s − 3.72·18-s + 0.174·19-s + 0.272·20-s − 0.438·21-s − 5.11·22-s + 4.72·23-s − 1.28·24-s − 3.75·25-s − 7.07·26-s − 2.51·27-s + 0.247·28-s + ⋯ |
L(s) = 1 | + 0.936·2-s + 0.250·3-s − 0.122·4-s − 0.498·5-s + 0.234·6-s − 0.382·7-s − 1.05·8-s − 0.937·9-s − 0.466·10-s − 1.16·11-s − 0.0306·12-s − 1.48·13-s − 0.358·14-s − 0.124·15-s − 0.862·16-s + 1.58·17-s − 0.878·18-s + 0.0399·19-s + 0.0610·20-s − 0.0956·21-s − 1.09·22-s + 0.984·23-s − 0.263·24-s − 0.751·25-s − 1.38·26-s − 0.484·27-s + 0.0468·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 619 | \( 1 + T \) |
good | 2 | \( 1 - 1.32T + 2T^{2} \) |
| 3 | \( 1 - 0.433T + 3T^{2} \) |
| 5 | \( 1 + 1.11T + 5T^{2} \) |
| 7 | \( 1 + 1.01T + 7T^{2} \) |
| 11 | \( 1 + 3.85T + 11T^{2} \) |
| 13 | \( 1 + 5.34T + 13T^{2} \) |
| 17 | \( 1 - 6.52T + 17T^{2} \) |
| 19 | \( 1 - 0.174T + 19T^{2} \) |
| 23 | \( 1 - 4.72T + 23T^{2} \) |
| 29 | \( 1 - 6.63T + 29T^{2} \) |
| 31 | \( 1 - 1.79T + 31T^{2} \) |
| 37 | \( 1 - 3.31T + 37T^{2} \) |
| 41 | \( 1 + 1.64T + 41T^{2} \) |
| 43 | \( 1 + 0.534T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 + 11.8T + 53T^{2} \) |
| 59 | \( 1 + 1.75T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 - 5.10T + 67T^{2} \) |
| 71 | \( 1 + 9.54T + 71T^{2} \) |
| 73 | \( 1 - 5.76T + 73T^{2} \) |
| 79 | \( 1 - 8.04T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 + 13.3T + 97T^{2} \) |
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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
−10.06414293244683573791800608124, −9.446343429852388846618501396065, −8.222653050907202998543997859681, −7.67387295214651135385956355518, −6.33359776496913207900171225340, −5.29711363508147175128727311369, −4.74198248094815234656620710356, −3.26056674085726468459806085525, −2.81109235008017523712803848187, 0,
2.81109235008017523712803848187, 3.26056674085726468459806085525, 4.74198248094815234656620710356, 5.29711363508147175128727311369, 6.33359776496913207900171225340, 7.67387295214651135385956355518, 8.222653050907202998543997859681, 9.446343429852388846618501396065, 10.06414293244683573791800608124