| L(s) = 1 | + 2-s − 1.48·3-s + 4-s + 5-s − 1.48·6-s − 3·7-s + 8-s − 0.785·9-s + 10-s − 11-s − 1.48·12-s + 4.05·13-s − 3·14-s − 1.48·15-s + 16-s + 6.71·17-s − 0.785·18-s − 5.68·19-s + 20-s + 4.46·21-s − 22-s + 0.790·23-s − 1.48·24-s + 25-s + 4.05·26-s + 5.63·27-s − 3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.859·3-s + 0.5·4-s + 0.447·5-s − 0.607·6-s − 1.13·7-s + 0.353·8-s − 0.261·9-s + 0.316·10-s − 0.301·11-s − 0.429·12-s + 1.12·13-s − 0.801·14-s − 0.384·15-s + 0.250·16-s + 1.62·17-s − 0.185·18-s − 1.30·19-s + 0.223·20-s + 0.974·21-s − 0.213·22-s + 0.164·23-s − 0.303·24-s + 0.200·25-s + 0.795·26-s + 1.08·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| good | 3 | \( 1 + 1.48T + 3T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 13 | \( 1 - 4.05T + 13T^{2} \) |
| 17 | \( 1 - 6.71T + 17T^{2} \) |
| 19 | \( 1 + 5.68T + 19T^{2} \) |
| 23 | \( 1 - 0.790T + 23T^{2} \) |
| 29 | \( 1 + 0.250T + 29T^{2} \) |
| 31 | \( 1 + 4.11T + 31T^{2} \) |
| 37 | \( 1 + 6.59T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 + 2.35T + 43T^{2} \) |
| 47 | \( 1 + 6.15T + 47T^{2} \) |
| 53 | \( 1 - 5.76T + 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 + 8.49T + 71T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + 0.265T + 83T^{2} \) |
| 89 | \( 1 - 8.56T + 89T^{2} \) |
| 97 | \( 1 - 7.00T + 97T^{2} \) |
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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
−7.01641989435128924425100233220, −6.64555884515704941478674590287, −5.94179170636110308537916161066, −5.53722273365726469662803847436, −4.93489002791820908987678479008, −3.68510315283618261535685365772, −3.39197018959142877180114824376, −2.36269477359805208419732159983, −1.24727639976758481196995493971, 0,
1.24727639976758481196995493971, 2.36269477359805208419732159983, 3.39197018959142877180114824376, 3.68510315283618261535685365772, 4.93489002791820908987678479008, 5.53722273365726469662803847436, 5.94179170636110308537916161066, 6.64555884515704941478674590287, 7.01641989435128924425100233220
