| L(s) = 1 | − 4.09e3·2-s − 1.36e6·3-s + 1.67e7·4-s + 5.58e9·6-s − 6.14e10·7-s − 6.87e10·8-s + 1.01e12·9-s + 1.16e13·11-s − 2.28e13·12-s + 5.44e13·13-s + 2.51e14·14-s + 2.81e14·16-s + 2.56e15·17-s − 4.13e15·18-s + 1.17e15·19-s + 8.37e16·21-s − 4.77e16·22-s − 3.81e16·23-s + 9.36e16·24-s − 2.22e17·26-s − 2.22e17·27-s − 1.03e18·28-s + 1.53e18·29-s + 6.41e18·31-s − 1.15e18·32-s − 1.58e19·33-s − 1.04e19·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.48·3-s + 0.5·4-s + 1.04·6-s − 1.67·7-s − 0.353·8-s + 1.19·9-s + 1.11·11-s − 0.740·12-s + 0.648·13-s + 1.18·14-s + 0.250·16-s + 1.06·17-s − 0.843·18-s + 0.122·19-s + 2.48·21-s − 0.791·22-s − 0.362·23-s + 0.523·24-s − 0.458·26-s − 0.285·27-s − 0.838·28-s + 0.806·29-s + 1.46·31-s − 0.176·32-s − 1.65·33-s − 0.754·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(\approx\) |
\(0.8523130555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8523130555\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4.09e3T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 1.36e6T + 8.47e11T^{2} \) |
| 7 | \( 1 + 6.14e10T + 1.34e21T^{2} \) |
| 11 | \( 1 - 1.16e13T + 1.08e26T^{2} \) |
| 13 | \( 1 - 5.44e13T + 7.05e27T^{2} \) |
| 17 | \( 1 - 2.56e15T + 5.77e30T^{2} \) |
| 19 | \( 1 - 1.17e15T + 9.30e31T^{2} \) |
| 23 | \( 1 + 3.81e16T + 1.10e34T^{2} \) |
| 29 | \( 1 - 1.53e18T + 3.63e36T^{2} \) |
| 31 | \( 1 - 6.41e18T + 1.92e37T^{2} \) |
| 37 | \( 1 + 4.08e19T + 1.60e39T^{2} \) |
| 41 | \( 1 - 2.38e20T + 2.08e40T^{2} \) |
| 43 | \( 1 - 3.46e20T + 6.86e40T^{2} \) |
| 47 | \( 1 - 1.30e21T + 6.34e41T^{2} \) |
| 53 | \( 1 - 1.36e21T + 1.27e43T^{2} \) |
| 59 | \( 1 - 2.63e21T + 1.86e44T^{2} \) |
| 61 | \( 1 + 3.46e22T + 4.29e44T^{2} \) |
| 67 | \( 1 + 2.01e22T + 4.48e45T^{2} \) |
| 71 | \( 1 + 1.27e23T + 1.91e46T^{2} \) |
| 73 | \( 1 + 3.22e23T + 3.82e46T^{2} \) |
| 79 | \( 1 - 8.05e23T + 2.75e47T^{2} \) |
| 83 | \( 1 - 7.78e23T + 9.48e47T^{2} \) |
| 89 | \( 1 + 4.04e24T + 5.42e48T^{2} \) |
| 97 | \( 1 + 9.27e24T + 4.66e49T^{2} \) |
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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
−10.70886665867342342847716401146, −9.934766558718177576533897426352, −8.952259024563190529989362509344, −7.25593881980319726955566019358, −6.24828375244506443769126337857, −5.89119984916218589800210601753, −4.13796511521541180040762241434, −2.94053348170299679686431244142, −1.13192203364311694255999075291, −0.57944673542836095061428808226,
0.57944673542836095061428808226, 1.13192203364311694255999075291, 2.94053348170299679686431244142, 4.13796511521541180040762241434, 5.89119984916218589800210601753, 6.24828375244506443769126337857, 7.25593881980319726955566019358, 8.952259024563190529989362509344, 9.934766558718177576533897426352, 10.70886665867342342847716401146
