| L(s) = 1 | + 1.27·2-s − 0.377·4-s − 3.65·7-s − 3.02·8-s − 2.65·11-s − 6.13·13-s − 4.65·14-s − 3.10·16-s + 2.34·17-s + 19-s − 3.37·22-s + 5.48·23-s − 7.81·26-s + 1.37·28-s − 0.651·29-s − 6.67·31-s + 2.10·32-s + 2.99·34-s + 8.70·37-s + 1.27·38-s − 1.93·41-s + 2.65·43-s + 0.999·44-s + 6.98·46-s + 3.71·47-s + 6.33·49-s + 2.31·52-s + ⋯ |
| L(s) = 1 | + 0.900·2-s − 0.188·4-s − 1.37·7-s − 1.07·8-s − 0.799·11-s − 1.70·13-s − 1.24·14-s − 0.775·16-s + 0.569·17-s + 0.229·19-s − 0.720·22-s + 1.14·23-s − 1.53·26-s + 0.260·28-s − 0.120·29-s − 1.19·31-s + 0.371·32-s + 0.513·34-s + 1.43·37-s + 0.206·38-s − 0.301·41-s + 0.405·43-s + 0.150·44-s + 1.02·46-s + 0.542·47-s + 0.904·49-s + 0.320·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.223471765\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.223471765\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 1.27T + 2T^{2} \) |
| 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
−8.430485169432328113012234996771, −7.30848830419549750562486742258, −7.01937045931268177550805129642, −5.85014867839437979829095057142, −5.45973485174787373467421880379, −4.68021651537084873304692788291, −3.82429276756085675053713750208, −2.96331046239774651272285782731, −2.51629820984934939324235352370, −0.51205925696299163643904050997,
0.51205925696299163643904050997, 2.51629820984934939324235352370, 2.96331046239774651272285782731, 3.82429276756085675053713750208, 4.68021651537084873304692788291, 5.45973485174787373467421880379, 5.85014867839437979829095057142, 7.01937045931268177550805129642, 7.30848830419549750562486742258, 8.430485169432328113012234996771
