| L(s) = 1 | + 1.53·2-s + 3-s + 0.347·4-s + 1.87·5-s + 1.53·6-s − 7-s − 2.53·8-s + 9-s + 2.87·10-s − 4.22·11-s + 0.347·12-s + 3.22·13-s − 1.53·14-s + 1.87·15-s − 4.57·16-s + 1.53·18-s − 2.41·19-s + 0.652·20-s − 21-s − 6.47·22-s − 5.63·23-s − 2.53·24-s − 1.46·25-s + 4.94·26-s + 27-s − 0.347·28-s − 1.36·29-s + ⋯ |
| L(s) = 1 | + 1.08·2-s + 0.577·3-s + 0.173·4-s + 0.840·5-s + 0.625·6-s − 0.377·7-s − 0.895·8-s + 0.333·9-s + 0.910·10-s − 1.27·11-s + 0.100·12-s + 0.894·13-s − 0.409·14-s + 0.485·15-s − 1.14·16-s + 0.361·18-s − 0.553·19-s + 0.145·20-s − 0.218·21-s − 1.38·22-s − 1.17·23-s − 0.516·24-s − 0.293·25-s + 0.969·26-s + 0.192·27-s − 0.0656·28-s − 0.254·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6069 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6069 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 1.53T + 2T^{2} \) |
| 5 | \( 1 - 1.87T + 5T^{2} \) |
| 11 | \( 1 + 4.22T + 11T^{2} \) |
| 13 | \( 1 - 3.22T + 13T^{2} \) |
| 19 | \( 1 + 2.41T + 19T^{2} \) |
| 23 | \( 1 + 5.63T + 23T^{2} \) |
| 29 | \( 1 + 1.36T + 29T^{2} \) |
| 31 | \( 1 + 4.75T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 + 0.751T + 41T^{2} \) |
| 43 | \( 1 + 8.86T + 43T^{2} \) |
| 47 | \( 1 + 8.33T + 47T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 59 | \( 1 - 7.31T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 - 9.58T + 67T^{2} \) |
| 71 | \( 1 - 11.5T + 71T^{2} \) |
| 73 | \( 1 + 12.5T + 73T^{2} \) |
| 79 | \( 1 + 7T + 79T^{2} \) |
| 83 | \( 1 - 2.83T + 83T^{2} \) |
| 89 | \( 1 - T + 89T^{2} \) |
| 97 | \( 1 + 6.08T + 97T^{2} \) |
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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
−7.87632114181274632166196086207, −6.68512217219889616381888843588, −6.10294190640759250837884667226, −5.58235862885513051973988012501, −4.81216672839611483814501721172, −3.99481768268861108201730518169, −3.30710086591974590997142623014, −2.54229364221570632756614677639, −1.76845286944158617798667783864, 0,
1.76845286944158617798667783864, 2.54229364221570632756614677639, 3.30710086591974590997142623014, 3.99481768268861108201730518169, 4.81216672839611483814501721172, 5.58235862885513051973988012501, 6.10294190640759250837884667226, 6.68512217219889616381888843588, 7.87632114181274632166196086207
