| L(s) = 1 | − 2.21·2-s + 3-s + 2.90·4-s − 2.21·6-s − 7-s − 2·8-s + 9-s − 11-s + 2.90·12-s + 1.31·13-s + 2.21·14-s − 1.37·16-s + 1.90·17-s − 2.21·18-s + 0.377·19-s − 21-s + 2.21·22-s + 2.52·23-s − 2·24-s − 2.90·26-s + 27-s − 2.90·28-s − 6.11·29-s − 4.96·31-s + 7.05·32-s − 33-s − 4.21·34-s + ⋯ |
| L(s) = 1 | − 1.56·2-s + 0.577·3-s + 1.45·4-s − 0.903·6-s − 0.377·7-s − 0.707·8-s + 0.333·9-s − 0.301·11-s + 0.838·12-s + 0.363·13-s + 0.591·14-s − 0.344·16-s + 0.461·17-s − 0.521·18-s + 0.0866·19-s − 0.218·21-s + 0.472·22-s + 0.526·23-s − 0.408·24-s − 0.569·26-s + 0.192·27-s − 0.548·28-s − 1.13·29-s − 0.892·31-s + 1.24·32-s − 0.174·33-s − 0.722·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 2.21T + 2T^{2} \) |
| 13 | \( 1 - 1.31T + 13T^{2} \) |
| 17 | \( 1 - 1.90T + 17T^{2} \) |
| 19 | \( 1 - 0.377T + 19T^{2} \) |
| 23 | \( 1 - 2.52T + 23T^{2} \) |
| 29 | \( 1 + 6.11T + 29T^{2} \) |
| 31 | \( 1 + 4.96T + 31T^{2} \) |
| 37 | \( 1 + 1.31T + 37T^{2} \) |
| 41 | \( 1 + 2.68T + 41T^{2} \) |
| 43 | \( 1 - 6.11T + 43T^{2} \) |
| 47 | \( 1 + 1.59T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 + 7.21T + 59T^{2} \) |
| 61 | \( 1 + 1.04T + 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 - 4.92T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 + 0.969T + 79T^{2} \) |
| 83 | \( 1 + 6.71T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 8.54T + 97T^{2} \) |
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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
−7.83638873750061708855271507206, −7.40325886345421783285210992188, −6.68903486782896512103234513853, −5.85457639757131026108094675159, −4.88660862869411635314381068457, −3.78744028809947340611431898113, −3.01026663467914679466163716483, −2.05941519658198001459521317807, −1.23128132943094283328492697775, 0,
1.23128132943094283328492697775, 2.05941519658198001459521317807, 3.01026663467914679466163716483, 3.78744028809947340611431898113, 4.88660862869411635314381068457, 5.85457639757131026108094675159, 6.68903486782896512103234513853, 7.40325886345421783285210992188, 7.83638873750061708855271507206
