| L(s) = 1 | − 2.70·2-s + 0.960·3-s + 5.32·4-s − 3.57·5-s − 2.59·6-s + 3.95·7-s − 8.99·8-s − 2.07·9-s + 9.66·10-s + 11-s + 5.11·12-s − 10.7·14-s − 3.43·15-s + 13.6·16-s + 2.79·17-s + 5.62·18-s − 0.00685·19-s − 19.0·20-s + 3.79·21-s − 2.70·22-s − 3.94·23-s − 8.63·24-s + 7.76·25-s − 4.87·27-s + 21.0·28-s + 6.09·29-s + 9.28·30-s + ⋯ |
| L(s) = 1 | − 1.91·2-s + 0.554·3-s + 2.66·4-s − 1.59·5-s − 1.06·6-s + 1.49·7-s − 3.17·8-s − 0.692·9-s + 3.05·10-s + 0.301·11-s + 1.47·12-s − 2.86·14-s − 0.885·15-s + 3.42·16-s + 0.678·17-s + 1.32·18-s − 0.00157·19-s − 4.25·20-s + 0.828·21-s − 0.576·22-s − 0.822·23-s − 1.76·24-s + 1.55·25-s − 0.938·27-s + 3.97·28-s + 1.13·29-s + 1.69·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6627584728\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6627584728\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.70T + 2T^{2} \) |
| 3 | \( 1 - 0.960T + 3T^{2} \) |
| 5 | \( 1 + 3.57T + 5T^{2} \) |
| 7 | \( 1 - 3.95T + 7T^{2} \) |
| 17 | \( 1 - 2.79T + 17T^{2} \) |
| 19 | \( 1 + 0.00685T + 19T^{2} \) |
| 23 | \( 1 + 3.94T + 23T^{2} \) |
| 29 | \( 1 - 6.09T + 29T^{2} \) |
| 31 | \( 1 - 0.242T + 31T^{2} \) |
| 37 | \( 1 - 0.801T + 37T^{2} \) |
| 41 | \( 1 + 6.58T + 41T^{2} \) |
| 43 | \( 1 + 6.24T + 43T^{2} \) |
| 47 | \( 1 - 6.29T + 47T^{2} \) |
| 53 | \( 1 - 9.02T + 53T^{2} \) |
| 59 | \( 1 - 4.00T + 59T^{2} \) |
| 61 | \( 1 + 6.32T + 61T^{2} \) |
| 67 | \( 1 - 8.97T + 67T^{2} \) |
| 71 | \( 1 - 8.03T + 71T^{2} \) |
| 73 | \( 1 - 0.166T + 73T^{2} \) |
| 79 | \( 1 + 9.30T + 79T^{2} \) |
| 83 | \( 1 - 0.547T + 83T^{2} \) |
| 89 | \( 1 - 17.0T + 89T^{2} \) |
| 97 | \( 1 - 8.78T + 97T^{2} \) |
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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
−8.786770801103578616579977723900, −8.449079650592316530558641053592, −7.925642233383616294443127180197, −7.51217453131576372278380881396, −6.56518750443513373311422714874, −5.29926624134864001405655422373, −3.98803729681005337270279840798, −2.99608103748976102430838532014, −1.90283266394824333722742977292, −0.71470258001110782496311380970,
0.71470258001110782496311380970, 1.90283266394824333722742977292, 2.99608103748976102430838532014, 3.98803729681005337270279840798, 5.29926624134864001405655422373, 6.56518750443513373311422714874, 7.51217453131576372278380881396, 7.925642233383616294443127180197, 8.449079650592316530558641053592, 8.786770801103578616579977723900
