| L(s) = 1 | + 0.301·2-s + 3-s − 1.90·4-s − 0.698·5-s + 0.301·6-s − 7-s − 1.17·8-s + 9-s − 0.210·10-s − 1.90·12-s + 4.43·13-s − 0.301·14-s − 0.698·15-s + 3.46·16-s − 6.80·17-s + 0.301·18-s + 1.78·19-s + 1.33·20-s − 21-s − 9.20·23-s − 1.17·24-s − 4.51·25-s + 1.33·26-s + 27-s + 1.90·28-s + 7.89·29-s − 0.210·30-s + ⋯ |
| L(s) = 1 | + 0.212·2-s + 0.577·3-s − 0.954·4-s − 0.312·5-s + 0.122·6-s − 0.377·7-s − 0.416·8-s + 0.333·9-s − 0.0665·10-s − 0.551·12-s + 1.22·13-s − 0.0804·14-s − 0.180·15-s + 0.866·16-s − 1.65·17-s + 0.0709·18-s + 0.410·19-s + 0.298·20-s − 0.218·21-s − 1.91·23-s − 0.240·24-s − 0.902·25-s + 0.261·26-s + 0.192·27-s + 0.360·28-s + 1.46·29-s − 0.0384·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.620121325\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.620121325\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.301T + 2T^{2} \) |
| 5 | \( 1 + 0.698T + 5T^{2} \) |
| 13 | \( 1 - 4.43T + 13T^{2} \) |
| 17 | \( 1 + 6.80T + 17T^{2} \) |
| 19 | \( 1 - 1.78T + 19T^{2} \) |
| 23 | \( 1 + 9.20T + 23T^{2} \) |
| 29 | \( 1 - 7.89T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 - 6.16T + 37T^{2} \) |
| 41 | \( 1 - 1.33T + 41T^{2} \) |
| 43 | \( 1 - 6.33T + 43T^{2} \) |
| 47 | \( 1 - 8.20T + 47T^{2} \) |
| 53 | \( 1 + 2.35T + 53T^{2} \) |
| 59 | \( 1 + 4.76T + 59T^{2} \) |
| 61 | \( 1 + 1.69T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 - 7.58T + 71T^{2} \) |
| 73 | \( 1 - 6.96T + 73T^{2} \) |
| 79 | \( 1 - 8.85T + 79T^{2} \) |
| 83 | \( 1 - 5.71T + 83T^{2} \) |
| 89 | \( 1 + 2.22T + 89T^{2} \) |
| 97 | \( 1 + 0.171T + 97T^{2} \) |
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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
−8.787019376917510027795223688127, −8.284780505524659854824244498913, −7.64465549042775252823978355855, −6.31040694133307296439764693047, −6.03070010377484614982457386292, −4.53956410924468408723668311126, −4.21318171166286894864072675440, −3.34674604456330356644430473746, −2.29994351920035005096452895197, −0.76279440149351108306993123535,
0.76279440149351108306993123535, 2.29994351920035005096452895197, 3.34674604456330356644430473746, 4.21318171166286894864072675440, 4.53956410924468408723668311126, 6.03070010377484614982457386292, 6.31040694133307296439764693047, 7.64465549042775252823978355855, 8.284780505524659854824244498913, 8.787019376917510027795223688127
