| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 3.23·7-s − 8-s + 9-s + 2·11-s + 12-s − 0.763·13-s + 3.23·14-s + 16-s − 17-s − 18-s + 2.47·19-s − 3.23·21-s − 2·22-s − 4·23-s − 24-s + 0.763·26-s + 27-s − 3.23·28-s − 4·29-s − 6.47·31-s − 32-s + 2·33-s + 34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s − 1.22·7-s − 0.353·8-s + 0.333·9-s + 0.603·11-s + 0.288·12-s − 0.211·13-s + 0.864·14-s + 0.250·16-s − 0.242·17-s − 0.235·18-s + 0.567·19-s − 0.706·21-s − 0.426·22-s − 0.834·23-s − 0.204·24-s + 0.149·26-s + 0.192·27-s − 0.611·28-s − 0.742·29-s − 1.16·31-s − 0.176·32-s + 0.348·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 3.23T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 0.763T + 13T^{2} \) |
| 19 | \( 1 - 2.47T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + 6.47T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 5.70T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 + 1.52T + 47T^{2} \) |
| 53 | \( 1 + 6.94T + 53T^{2} \) |
| 59 | \( 1 + 1.70T + 59T^{2} \) |
| 61 | \( 1 + 4.47T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 - 6.76T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 + 16.9T + 79T^{2} \) |
| 83 | \( 1 + 1.52T + 83T^{2} \) |
| 89 | \( 1 - 3.52T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−8.752677768457262890497191805121, −7.66829346390298021077226010320, −7.23257998762657334452949963470, −6.32745162273406458023427021759, −5.68829218510361101543219965577, −4.26921732194890432187040349064, −3.45854097395211246013378726282, −2.63839662389258357743615785747, −1.52521759800200905206610807774, 0,
1.52521759800200905206610807774, 2.63839662389258357743615785747, 3.45854097395211246013378726282, 4.26921732194890432187040349064, 5.68829218510361101543219965577, 6.32745162273406458023427021759, 7.23257998762657334452949963470, 7.66829346390298021077226010320, 8.752677768457262890497191805121
