| L(s) = 1 | − 2.07·2-s − 1.13·3-s + 2.31·4-s − 4.26·5-s + 2.35·6-s − 3.46·7-s − 0.655·8-s − 1.70·9-s + 8.86·10-s + 11-s − 2.63·12-s + 7.19·14-s + 4.84·15-s − 3.26·16-s + 3.63·17-s + 3.55·18-s − 4.71·19-s − 9.88·20-s + 3.93·21-s − 2.07·22-s − 1.92·23-s + 0.745·24-s + 13.2·25-s + 5.34·27-s − 8.01·28-s − 1.02·29-s − 10.0·30-s + ⋯ |
| L(s) = 1 | − 1.46·2-s − 0.655·3-s + 1.15·4-s − 1.90·5-s + 0.963·6-s − 1.30·7-s − 0.231·8-s − 0.569·9-s + 2.80·10-s + 0.301·11-s − 0.759·12-s + 1.92·14-s + 1.25·15-s − 0.817·16-s + 0.880·17-s + 0.837·18-s − 1.08·19-s − 2.21·20-s + 0.858·21-s − 0.442·22-s − 0.400·23-s + 0.152·24-s + 2.64·25-s + 1.02·27-s − 1.51·28-s − 0.189·29-s − 1.83·30-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.07T + 2T^{2} \) |
| 3 | \( 1 + 1.13T + 3T^{2} \) |
| 5 | \( 1 + 4.26T + 5T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 17 | \( 1 - 3.63T + 17T^{2} \) |
| 19 | \( 1 + 4.71T + 19T^{2} \) |
| 23 | \( 1 + 1.92T + 23T^{2} \) |
| 29 | \( 1 + 1.02T + 29T^{2} \) |
| 31 | \( 1 - 5.79T + 31T^{2} \) |
| 37 | \( 1 + 3.78T + 37T^{2} \) |
| 41 | \( 1 - 6.78T + 41T^{2} \) |
| 43 | \( 1 - 1.80T + 43T^{2} \) |
| 47 | \( 1 - 7.76T + 47T^{2} \) |
| 53 | \( 1 - 2.40T + 53T^{2} \) |
| 59 | \( 1 + 2.64T + 59T^{2} \) |
| 61 | \( 1 + 9.38T + 61T^{2} \) |
| 67 | \( 1 - 3.52T + 67T^{2} \) |
| 71 | \( 1 + 5.25T + 71T^{2} \) |
| 73 | \( 1 + 6.31T + 73T^{2} \) |
| 79 | \( 1 - 6.50T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 - 8.08T + 89T^{2} \) |
| 97 | \( 1 + 9.42T + 97T^{2} \) |
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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
−8.784073951499446552506525834624, −8.147007105403868399248457696049, −7.47071224928858799637354868080, −6.72504187950834217883270231568, −6.01436156296900100738062683477, −4.60528903949956675756046606614, −3.72789555917895968952012201707, −2.76098040194277056418774361940, −0.796810832862870638666661236839, 0,
0.796810832862870638666661236839, 2.76098040194277056418774361940, 3.72789555917895968952012201707, 4.60528903949956675756046606614, 6.01436156296900100738062683477, 6.72504187950834217883270231568, 7.47071224928858799637354868080, 8.147007105403868399248457696049, 8.784073951499446552506525834624
