| L(s) = 1 | + 2.41·2-s + 0.585·3-s + 3.82·4-s + 1.41·6-s + 3.41·7-s + 4.41·8-s − 2.65·9-s − 5.41·11-s + 2.24·12-s − 2.82·13-s + 8.24·14-s + 2.99·16-s + 17-s − 6.41·18-s + 2.82·19-s + 2·21-s − 13.0·22-s + 0.585·23-s + 2.58·24-s − 6.82·26-s − 3.31·27-s + 13.0·28-s + 0.828·29-s − 4.24·31-s − 1.58·32-s − 3.17·33-s + 2.41·34-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 0.338·3-s + 1.91·4-s + 0.577·6-s + 1.29·7-s + 1.56·8-s − 0.885·9-s − 1.63·11-s + 0.647·12-s − 0.784·13-s + 2.20·14-s + 0.749·16-s + 0.242·17-s − 1.51·18-s + 0.648·19-s + 0.436·21-s − 2.78·22-s + 0.122·23-s + 0.527·24-s − 1.33·26-s − 0.637·27-s + 2.47·28-s + 0.153·29-s − 0.762·31-s − 0.280·32-s − 0.552·33-s + 0.414·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.742523863\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.742523863\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 3 | \( 1 - 0.585T + 3T^{2} \) |
| 7 | \( 1 - 3.41T + 7T^{2} \) |
| 11 | \( 1 + 5.41T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 0.585T + 23T^{2} \) |
| 29 | \( 1 - 0.828T + 29T^{2} \) |
| 31 | \( 1 + 4.24T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 - 3.65T + 43T^{2} \) |
| 47 | \( 1 + 0.828T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 + 14.8T + 59T^{2} \) |
| 61 | \( 1 + 3.65T + 61T^{2} \) |
| 67 | \( 1 - 8.82T + 67T^{2} \) |
| 71 | \( 1 - 4.24T + 71T^{2} \) |
| 73 | \( 1 + 0.828T + 73T^{2} \) |
| 79 | \( 1 - 2.58T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 - 7.65T + 97T^{2} \) |
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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
−11.23664520772967716513190019753, −10.90577701985937228495745063114, −9.411322287712602102775235249481, −7.931599288782309644467158872189, −7.58492659120522843879784448101, −5.95927652893199961786478029183, −5.21187388188051352096779670999, −4.55161728048305662732947370380, −3.06745154060573483765755880299, −2.28272312873032065049441638696,
2.28272312873032065049441638696, 3.06745154060573483765755880299, 4.55161728048305662732947370380, 5.21187388188051352096779670999, 5.95927652893199961786478029183, 7.58492659120522843879784448101, 7.931599288782309644467158872189, 9.411322287712602102775235249481, 10.90577701985937228495745063114, 11.23664520772967716513190019753
