| L(s) = 1 | + 1.25e18·2-s − 1.09e28·3-s + 9.17e35·4-s − 6.18e41·5-s − 1.37e46·6-s − 3.09e50·7-s + 3.18e53·8-s − 4.78e56·9-s − 7.77e59·10-s + 1.28e62·11-s − 1.00e64·12-s − 7.15e65·13-s − 3.88e68·14-s + 6.77e69·15-s − 2.09e71·16-s − 1.37e73·17-s − 6.02e74·18-s + 1.31e75·19-s − 5.67e77·20-s + 3.38e78·21-s + 1.61e80·22-s + 1.25e81·23-s − 3.48e81·24-s + 2.31e83·25-s − 9.00e83·26-s + 1.18e85·27-s − 2.83e86·28-s + ⋯ |
| L(s) = 1 | + 1.54·2-s − 0.448·3-s + 1.38·4-s − 1.59·5-s − 0.691·6-s − 1.60·7-s + 0.587·8-s − 0.799·9-s − 2.45·10-s + 1.40·11-s − 0.618·12-s − 0.376·13-s − 2.48·14-s + 0.714·15-s − 0.474·16-s − 0.846·17-s − 1.23·18-s + 0.107·19-s − 2.20·20-s + 0.721·21-s + 2.16·22-s + 1.18·23-s − 0.263·24-s + 1.54·25-s − 0.580·26-s + 0.806·27-s − 2.22·28-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
| \(L(60)\) |
\(\approx\) |
\(1.432606762\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.432606762\) |
| \(L(\frac{121}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 1.25e18T + 6.64e35T^{2} \) |
| 3 | \( 1 + 1.09e28T + 5.99e56T^{2} \) |
| 5 | \( 1 + 6.18e41T + 1.50e83T^{2} \) |
| 7 | \( 1 + 3.09e50T + 3.68e100T^{2} \) |
| 11 | \( 1 - 1.28e62T + 8.42e123T^{2} \) |
| 13 | \( 1 + 7.15e65T + 3.62e132T^{2} \) |
| 17 | \( 1 + 1.37e73T + 2.65e146T^{2} \) |
| 19 | \( 1 - 1.31e75T + 1.48e152T^{2} \) |
| 23 | \( 1 - 1.25e81T + 1.11e162T^{2} \) |
| 29 | \( 1 + 5.10e86T + 1.06e174T^{2} \) |
| 31 | \( 1 - 3.03e88T + 2.96e177T^{2} \) |
| 37 | \( 1 - 9.96e92T + 4.13e186T^{2} \) |
| 41 | \( 1 - 9.49e95T + 8.34e191T^{2} \) |
| 43 | \( 1 + 2.68e97T + 2.41e194T^{2} \) |
| 47 | \( 1 - 1.25e99T + 9.54e198T^{2} \) |
| 53 | \( 1 + 5.87e102T + 1.54e205T^{2} \) |
| 59 | \( 1 - 2.33e105T + 5.38e210T^{2} \) |
| 61 | \( 1 + 5.10e104T + 2.84e212T^{2} \) |
| 67 | \( 1 + 4.46e108T + 2.00e217T^{2} \) |
| 71 | \( 1 + 8.71e109T + 1.99e220T^{2} \) |
| 73 | \( 1 - 1.10e110T + 5.43e221T^{2} \) |
| 79 | \( 1 - 6.92e112T + 6.57e225T^{2} \) |
| 83 | \( 1 + 2.69e114T + 2.34e228T^{2} \) |
| 89 | \( 1 - 9.23e115T + 9.49e231T^{2} \) |
| 97 | \( 1 - 1.37e118T + 2.66e236T^{2} \) |
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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
−13.02970094932542034215514467838, −12.00015833766387887418477311658, −11.29114829799525257272918968409, −9.028309358739167315646327655367, −6.97277700813990431814912230780, −6.19638761194364520933600136228, −4.63995518303389861949633064943, −3.64020306211102176875723226318, −2.92425327545714489339894377393, −0.45451799362246062581743690918,
0.45451799362246062581743690918, 2.92425327545714489339894377393, 3.64020306211102176875723226318, 4.63995518303389861949633064943, 6.19638761194364520933600136228, 6.97277700813990431814912230780, 9.028309358739167315646327655367, 11.29114829799525257272918968409, 12.00015833766387887418477311658, 13.02970094932542034215514467838
