| L(s) = 1 | + 1.27·2-s − 1.65·3-s − 0.377·4-s − 2.10·6-s + 3.65·7-s − 3.02·8-s − 0.273·9-s + 2.65·11-s + 0.622·12-s + 6.13·13-s + 4.65·14-s − 3.10·16-s + 2.34·17-s − 0.348·18-s + 19-s − 6.02·21-s + 3.37·22-s + 5.48·23-s + 5·24-s + 7.81·26-s + 5.40·27-s − 1.37·28-s + 0.651·29-s − 6.67·31-s + 2.10·32-s − 4.37·33-s + 2.99·34-s + ⋯ |
| L(s) = 1 | + 0.900·2-s − 0.953·3-s − 0.188·4-s − 0.858·6-s + 1.37·7-s − 1.07·8-s − 0.0912·9-s + 0.799·11-s + 0.179·12-s + 1.70·13-s + 1.24·14-s − 0.775·16-s + 0.569·17-s − 0.0822·18-s + 0.229·19-s − 1.31·21-s + 0.720·22-s + 1.14·23-s + 1.02·24-s + 1.53·26-s + 1.04·27-s − 0.260·28-s + 0.120·29-s − 1.19·31-s + 0.371·32-s − 0.761·33-s + 0.513·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.682612882\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.682612882\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 1.27T + 2T^{2} \) |
| 3 | \( 1 + 1.65T + 3T^{2} \) |
| 7 | \( 1 - 3.65T + 7T^{2} \) |
| 11 | \( 1 - 2.65T + 11T^{2} \) |
| 13 | \( 1 - 6.13T + 13T^{2} \) |
| 17 | \( 1 - 2.34T + 17T^{2} \) |
| 23 | \( 1 - 5.48T + 23T^{2} \) |
| 29 | \( 1 - 0.651T + 29T^{2} \) |
| 31 | \( 1 + 6.67T + 31T^{2} \) |
| 37 | \( 1 + 8.70T + 37T^{2} \) |
| 41 | \( 1 - 1.93T + 41T^{2} \) |
| 43 | \( 1 + 2.65T + 43T^{2} \) |
| 47 | \( 1 - 3.71T + 47T^{2} \) |
| 53 | \( 1 - 13.7T + 53T^{2} \) |
| 59 | \( 1 + 7.84T + 59T^{2} \) |
| 61 | \( 1 + 1.92T + 61T^{2} \) |
| 67 | \( 1 + 4.44T + 67T^{2} \) |
| 71 | \( 1 - 3.54T + 71T^{2} \) |
| 73 | \( 1 + 2.48T + 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 + 5.06T + 89T^{2} \) |
| 97 | \( 1 - 3.22T + 97T^{2} \) |
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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
−11.31905151467784379765746494530, −10.55306986886615390048065866020, −8.971389311076073385510240876919, −8.503499574165846804232151609551, −7.03739220695622120410663572853, −5.88465443674296909498651136912, −5.38259941487293628390199238209, −4.41656585771383734513287164784, −3.39243230360879815973047494418, −1.25788172204287998021380646619,
1.25788172204287998021380646619, 3.39243230360879815973047494418, 4.41656585771383734513287164784, 5.38259941487293628390199238209, 5.88465443674296909498651136912, 7.03739220695622120410663572853, 8.503499574165846804232151609551, 8.971389311076073385510240876919, 10.55306986886615390048065866020, 11.31905151467784379765746494530
