| L(s) = 1 | + 1.33·2-s + 3.35·3-s − 0.216·4-s − 5-s + 4.47·6-s − 2.95·8-s + 8.24·9-s − 1.33·10-s − 5.20·11-s − 0.724·12-s + 2.72·13-s − 3.35·15-s − 3.52·16-s − 6.47·17-s + 11.0·18-s − 2.69·19-s + 0.216·20-s − 6.94·22-s − 3.79·23-s − 9.92·24-s + 25-s + 3.63·26-s + 17.5·27-s + 29-s − 4.47·30-s − 9.20·31-s + 1.21·32-s + ⋯ |
| L(s) = 1 | + 0.944·2-s + 1.93·3-s − 0.108·4-s − 0.447·5-s + 1.82·6-s − 1.04·8-s + 2.74·9-s − 0.422·10-s − 1.56·11-s − 0.209·12-s + 0.754·13-s − 0.865·15-s − 0.880·16-s − 1.57·17-s + 2.59·18-s − 0.618·19-s + 0.0483·20-s − 1.48·22-s − 0.791·23-s − 2.02·24-s + 0.200·25-s + 0.712·26-s + 3.38·27-s + 0.185·29-s − 0.817·30-s − 1.65·31-s + 0.215·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 1.33T + 2T^{2} \) |
| 3 | \( 1 - 3.35T + 3T^{2} \) |
| 11 | \( 1 + 5.20T + 11T^{2} \) |
| 13 | \( 1 - 2.72T + 13T^{2} \) |
| 17 | \( 1 + 6.47T + 17T^{2} \) |
| 19 | \( 1 + 2.69T + 19T^{2} \) |
| 23 | \( 1 + 3.79T + 23T^{2} \) |
| 31 | \( 1 + 9.20T + 31T^{2} \) |
| 37 | \( 1 - 1.45T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 - 2.24T + 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 - 5.24T + 53T^{2} \) |
| 59 | \( 1 + 1.14T + 59T^{2} \) |
| 61 | \( 1 + 2.07T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 - 5.96T + 71T^{2} \) |
| 73 | \( 1 - 3.92T + 73T^{2} \) |
| 79 | \( 1 - 8.37T + 79T^{2} \) |
| 83 | \( 1 - 8.40T + 83T^{2} \) |
| 89 | \( 1 - 12.0T + 89T^{2} \) |
| 97 | \( 1 + 1.87T + 97T^{2} \) |
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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
−7.84658972812243809239216662472, −6.94267653849433904951626687209, −6.24986328202960157358611268833, −5.08390177108403312010620855750, −4.54801166019587440374558159242, −3.77611810550912573435541833680, −3.36312535569617953731643737377, −2.49789389017988314942539595668, −1.90606153369517402767992457590, 0,
1.90606153369517402767992457590, 2.49789389017988314942539595668, 3.36312535569617953731643737377, 3.77611810550912573435541833680, 4.54801166019587440374558159242, 5.08390177108403312010620855750, 6.24986328202960157358611268833, 6.94267653849433904951626687209, 7.84658972812243809239216662472
