| L(s) = 1 | + 1.61·2-s − 3-s + 0.618·4-s − 1.61·6-s + 5.23·7-s − 2.23·8-s + 9-s − 0.618·12-s − 3.23·13-s + 8.47·14-s − 4.85·16-s − 2·17-s + 1.61·18-s − 5·19-s − 5.23·21-s + 4.61·23-s + 2.23·24-s − 5.23·26-s − 27-s + 3.23·28-s − 0.854·29-s + 7·31-s − 3.38·32-s − 3.23·34-s + 0.618·36-s + 1.47·37-s − 8.09·38-s + ⋯ |
| L(s) = 1 | + 1.14·2-s − 0.577·3-s + 0.309·4-s − 0.660·6-s + 1.97·7-s − 0.790·8-s + 0.333·9-s − 0.178·12-s − 0.897·13-s + 2.26·14-s − 1.21·16-s − 0.485·17-s + 0.381·18-s − 1.14·19-s − 1.14·21-s + 0.962·23-s + 0.456·24-s − 1.02·26-s − 0.192·27-s + 0.611·28-s − 0.158·29-s + 1.25·31-s − 0.597·32-s − 0.554·34-s + 0.103·36-s + 0.242·37-s − 1.31·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 7 | \( 1 - 5.23T + 7T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 - 4.61T + 23T^{2} \) |
| 29 | \( 1 + 0.854T + 29T^{2} \) |
| 31 | \( 1 - 7T + 31T^{2} \) |
| 37 | \( 1 - 1.47T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 + 9.09T + 43T^{2} \) |
| 47 | \( 1 + 6.09T + 47T^{2} \) |
| 53 | \( 1 + 5.38T + 53T^{2} \) |
| 59 | \( 1 - 6.70T + 59T^{2} \) |
| 61 | \( 1 + 7T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 4.52T + 73T^{2} \) |
| 79 | \( 1 + 3.09T + 79T^{2} \) |
| 83 | \( 1 + 7.70T + 83T^{2} \) |
| 89 | \( 1 - 4.14T + 89T^{2} \) |
| 97 | \( 1 - 2.52T + 97T^{2} \) |
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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
−7.11800624821400551754289123052, −6.61600411412818746505104908886, −5.74534023138898038564974919006, −5.05655618736389795039405526814, −4.65447304105399155249505213361, −4.34350257298173795656156987886, −3.18028882708819551868685773777, −2.24885001013049500908693587802, −1.44966101573753797828466744749, 0,
1.44966101573753797828466744749, 2.24885001013049500908693587802, 3.18028882708819551868685773777, 4.34350257298173795656156987886, 4.65447304105399155249505213361, 5.05655618736389795039405526814, 5.74534023138898038564974919006, 6.61600411412818746505104908886, 7.11800624821400551754289123052
