| L(s) = 1 | − 2.81·2-s − 3-s + 5.91·4-s − 1.28·5-s + 2.81·6-s − 7-s − 11.0·8-s + 9-s + 3.62·10-s + 4.20·11-s − 5.91·12-s − 13-s + 2.81·14-s + 1.28·15-s + 19.1·16-s + 1.62·17-s − 2.81·18-s − 6.33·19-s − 7.62·20-s + 21-s − 11.8·22-s − 2.71·23-s + 11.0·24-s − 3.33·25-s + 2.81·26-s − 27-s − 5.91·28-s + ⋯ |
| L(s) = 1 | − 1.98·2-s − 0.577·3-s + 2.95·4-s − 0.576·5-s + 1.14·6-s − 0.377·7-s − 3.89·8-s + 0.333·9-s + 1.14·10-s + 1.26·11-s − 1.70·12-s − 0.277·13-s + 0.751·14-s + 0.332·15-s + 4.79·16-s + 0.394·17-s − 0.663·18-s − 1.45·19-s − 1.70·20-s + 0.218·21-s − 2.52·22-s − 0.565·23-s + 2.24·24-s − 0.667·25-s + 0.551·26-s − 0.192·27-s − 1.11·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 2.81T + 2T^{2} \) |
| 5 | \( 1 + 1.28T + 5T^{2} \) |
| 11 | \( 1 - 4.20T + 11T^{2} \) |
| 17 | \( 1 - 1.62T + 17T^{2} \) |
| 19 | \( 1 + 6.33T + 19T^{2} \) |
| 23 | \( 1 + 2.71T + 23T^{2} \) |
| 29 | \( 1 + 4.33T + 29T^{2} \) |
| 31 | \( 1 + 7.49T + 31T^{2} \) |
| 37 | \( 1 - 3.42T + 37T^{2} \) |
| 41 | \( 1 + 7.62T + 41T^{2} \) |
| 43 | \( 1 + 4.91T + 43T^{2} \) |
| 47 | \( 1 + 1.08T + 47T^{2} \) |
| 53 | \( 1 - 1.75T + 53T^{2} \) |
| 59 | \( 1 + 4.57T + 59T^{2} \) |
| 61 | \( 1 + 5.25T + 61T^{2} \) |
| 67 | \( 1 - 8.67T + 67T^{2} \) |
| 71 | \( 1 + 6.78T + 71T^{2} \) |
| 73 | \( 1 - 4.07T + 73T^{2} \) |
| 79 | \( 1 + 8.91T + 79T^{2} \) |
| 83 | \( 1 + 4.33T + 83T^{2} \) |
| 89 | \( 1 - 4.54T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−11.27750660094113438982769717444, −10.32685444937641283860534458383, −9.516730262353179719824907430644, −8.664501218032730731758554958692, −7.64935459807238590060030704132, −6.76895350317749116357626020143, −5.96638571999111979418595354651, −3.68583120448531760208184043786, −1.80095072107294184984518087908, 0,
1.80095072107294184984518087908, 3.68583120448531760208184043786, 5.96638571999111979418595354651, 6.76895350317749116357626020143, 7.64935459807238590060030704132, 8.664501218032730731758554958692, 9.516730262353179719824907430644, 10.32685444937641283860534458383, 11.27750660094113438982769717444
