| L(s) = 1 | − 5-s − 1.29·7-s + 11-s − 5.01·13-s − 3.29·17-s − 5.71·19-s − 5.71·23-s + 25-s − 2·29-s − 3.71·31-s + 1.29·35-s + 2·37-s − 2·41-s + 4.41·43-s + 9.71·47-s − 5.31·49-s − 4.59·53-s − 55-s − 3.71·59-s + 15.1·61-s + 5.01·65-s + 1.40·67-s + 16.0·71-s − 5.89·73-s − 1.29·77-s + 14.0·79-s + 7.52·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.491·7-s + 0.301·11-s − 1.38·13-s − 0.800·17-s − 1.31·19-s − 1.19·23-s + 0.200·25-s − 0.371·29-s − 0.666·31-s + 0.219·35-s + 0.328·37-s − 0.312·41-s + 0.672·43-s + 1.41·47-s − 0.758·49-s − 0.631·53-s − 0.134·55-s − 0.483·59-s + 1.93·61-s + 0.621·65-s + 0.171·67-s + 1.90·71-s − 0.690·73-s − 0.148·77-s + 1.57·79-s + 0.825·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7611841878\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7611841878\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 + 1.29T + 7T^{2} \) |
| 13 | \( 1 + 5.01T + 13T^{2} \) |
| 17 | \( 1 + 3.29T + 17T^{2} \) |
| 19 | \( 1 + 5.71T + 19T^{2} \) |
| 23 | \( 1 + 5.71T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 3.71T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 4.41T + 43T^{2} \) |
| 47 | \( 1 - 9.71T + 47T^{2} \) |
| 53 | \( 1 + 4.59T + 53T^{2} \) |
| 59 | \( 1 + 3.71T + 59T^{2} \) |
| 61 | \( 1 - 15.1T + 61T^{2} \) |
| 67 | \( 1 - 1.40T + 67T^{2} \) |
| 71 | \( 1 - 16.0T + 71T^{2} \) |
| 73 | \( 1 + 5.89T + 73T^{2} \) |
| 79 | \( 1 - 14.0T + 79T^{2} \) |
| 83 | \( 1 - 7.52T + 83T^{2} \) |
| 89 | \( 1 + 16.5T + 89T^{2} \) |
| 97 | \( 1 + 6.82T + 97T^{2} \) |
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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
−7.85078057397479306078722521104, −7.08588538246036496353085511327, −6.57565914071588866255000517095, −5.83294363793922668496261532781, −4.94769682880040317224161619617, −4.21440329507926158668823124887, −3.67989048443947794201973746108, −2.53773993258718925036281094637, −1.99230321558861440120664656278, −0.40568118235162382630963376134,
0.40568118235162382630963376134, 1.99230321558861440120664656278, 2.53773993258718925036281094637, 3.67989048443947794201973746108, 4.21440329507926158668823124887, 4.94769682880040317224161619617, 5.83294363793922668496261532781, 6.57565914071588866255000517095, 7.08588538246036496353085511327, 7.85078057397479306078722521104
