| L(s) = 1 | + 1.04e18·2-s − 4.57e28·3-s + 4.19e35·4-s + 5.48e41·5-s − 4.76e46·6-s + 1.41e50·7-s − 2.54e53·8-s + 1.49e57·9-s + 5.70e59·10-s + 9.38e61·11-s − 1.92e64·12-s + 7.20e65·13-s + 1.47e68·14-s − 2.50e70·15-s − 5.44e71·16-s − 2.00e73·17-s + 1.55e75·18-s + 1.08e76·19-s + 2.30e77·20-s − 6.49e78·21-s + 9.77e79·22-s − 7.69e80·23-s + 1.16e82·24-s + 1.50e83·25-s + 7.50e83·26-s − 4.10e85·27-s + 5.96e85·28-s + ⋯ |
| L(s) = 1 | + 1.27·2-s − 1.86·3-s + 0.631·4-s + 1.41·5-s − 2.38·6-s + 0.739·7-s − 0.470·8-s + 2.49·9-s + 1.80·10-s + 1.02·11-s − 1.18·12-s + 0.378·13-s + 0.944·14-s − 2.64·15-s − 1.23·16-s − 1.23·17-s + 3.18·18-s + 0.893·19-s + 0.892·20-s − 1.38·21-s + 1.30·22-s − 0.730·23-s + 0.879·24-s + 0.997·25-s + 0.483·26-s − 2.79·27-s + 0.467·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(120-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+59.5) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(60)\) |
\(\approx\) |
\(3.392429299\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.392429299\) |
| \(L(\frac{121}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 1.04e18T + 6.64e35T^{2} \) |
| 3 | \( 1 + 4.57e28T + 5.99e56T^{2} \) |
| 5 | \( 1 - 5.48e41T + 1.50e83T^{2} \) |
| 7 | \( 1 - 1.41e50T + 3.68e100T^{2} \) |
| 11 | \( 1 - 9.38e61T + 8.42e123T^{2} \) |
| 13 | \( 1 - 7.20e65T + 3.62e132T^{2} \) |
| 17 | \( 1 + 2.00e73T + 2.65e146T^{2} \) |
| 19 | \( 1 - 1.08e76T + 1.48e152T^{2} \) |
| 23 | \( 1 + 7.69e80T + 1.11e162T^{2} \) |
| 29 | \( 1 - 8.23e86T + 1.06e174T^{2} \) |
| 31 | \( 1 + 1.15e88T + 2.96e177T^{2} \) |
| 37 | \( 1 - 8.65e92T + 4.13e186T^{2} \) |
| 41 | \( 1 - 3.98e95T + 8.34e191T^{2} \) |
| 43 | \( 1 - 2.71e97T + 2.41e194T^{2} \) |
| 47 | \( 1 + 1.18e99T + 9.54e198T^{2} \) |
| 53 | \( 1 + 3.63e102T + 1.54e205T^{2} \) |
| 59 | \( 1 - 3.27e105T + 5.38e210T^{2} \) |
| 61 | \( 1 - 2.04e106T + 2.84e212T^{2} \) |
| 67 | \( 1 + 4.46e108T + 2.00e217T^{2} \) |
| 71 | \( 1 - 1.37e110T + 1.99e220T^{2} \) |
| 73 | \( 1 - 2.52e110T + 5.43e221T^{2} \) |
| 79 | \( 1 - 1.99e112T + 6.57e225T^{2} \) |
| 83 | \( 1 - 1.95e114T + 2.34e228T^{2} \) |
| 89 | \( 1 - 8.02e115T + 9.49e231T^{2} \) |
| 97 | \( 1 - 5.97e117T + 2.66e236T^{2} \) |
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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
−13.15126091770709803855047358684, −11.93698926793340303050299182289, −11.00298135137405108023351168551, −9.480154434686120639686603237173, −6.62149730450693961360917556726, −5.96392503697026825098350162334, −5.08704214487613141565984708683, −4.16987223723524955840638755989, −2.00134576064234203094420233975, −0.868711638829783313400975511870,
0.868711638829783313400975511870, 2.00134576064234203094420233975, 4.16987223723524955840638755989, 5.08704214487613141565984708683, 5.96392503697026825098350162334, 6.62149730450693961360917556726, 9.480154434686120639686603237173, 11.00298135137405108023351168551, 11.93698926793340303050299182289, 13.15126091770709803855047358684
