L(s) = 1 | − 2-s − 3-s + 4-s + 3.06·5-s + 6-s − 0.702·7-s − 8-s + 9-s − 3.06·10-s + 1.81·11-s − 12-s − 6.14·13-s + 0.702·14-s − 3.06·15-s + 16-s − 3.70·17-s − 18-s − 19-s + 3.06·20-s + 0.702·21-s − 1.81·22-s − 6.68·23-s + 24-s + 4.39·25-s + 6.14·26-s − 27-s − 0.702·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.37·5-s + 0.408·6-s − 0.265·7-s − 0.353·8-s + 0.333·9-s − 0.969·10-s + 0.545·11-s − 0.288·12-s − 1.70·13-s + 0.187·14-s − 0.791·15-s + 0.250·16-s − 0.899·17-s − 0.235·18-s − 0.229·19-s + 0.685·20-s + 0.153·21-s − 0.385·22-s − 1.39·23-s + 0.204·24-s + 0.879·25-s + 1.20·26-s − 0.192·27-s − 0.132·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 + T \) |
good | 5 | \( 1 - 3.06T + 5T^{2} \) |
| 7 | \( 1 + 0.702T + 7T^{2} \) |
| 11 | \( 1 - 1.81T + 11T^{2} \) |
| 13 | \( 1 + 6.14T + 13T^{2} \) |
| 17 | \( 1 + 3.70T + 17T^{2} \) |
| 23 | \( 1 + 6.68T + 23T^{2} \) |
| 29 | \( 1 - 8.71T + 29T^{2} \) |
| 31 | \( 1 - 3.96T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 - 3.89T + 41T^{2} \) |
| 43 | \( 1 + 4.71T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 59 | \( 1 + 8.72T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 - 3.97T + 67T^{2} \) |
| 71 | \( 1 - 6.35T + 71T^{2} \) |
| 73 | \( 1 + 13.0T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 + 4.59T + 83T^{2} \) |
| 89 | \( 1 - 9.95T + 89T^{2} \) |
| 97 | \( 1 - 4.27T + 97T^{2} \) |
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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
−7.68096229274149679545458545406, −6.90978390929292760625738240608, −6.20938597090621165402310291926, −5.97504165762023272752525657076, −4.85110097022432994969030936428, −4.30307685624333727841934260259, −2.73461669617673815882263532283, −2.26605108027438659842603369028, −1.26830118886571971243176902609, 0,
1.26830118886571971243176902609, 2.26605108027438659842603369028, 2.73461669617673815882263532283, 4.30307685624333727841934260259, 4.85110097022432994969030936428, 5.97504165762023272752525657076, 6.20938597090621165402310291926, 6.90978390929292760625738240608, 7.68096229274149679545458545406