| L(s) = 1 | − 0.532·5-s − 7-s − 2.22·11-s + 2.06·13-s − 0.815·17-s + 7.94·19-s + 6.80·23-s − 4.71·25-s + 7.47·29-s − 2.29·31-s + 0.532·35-s − 10.2·37-s + 2.59·41-s − 0.290·43-s + 0.426·47-s + 49-s − 1.41·53-s + 1.18·55-s + 3.43·59-s − 10.4·61-s − 1.09·65-s + 4.20·67-s + 12.0·71-s − 9.09·73-s + 2.22·77-s − 7.46·79-s − 17.5·83-s + ⋯ |
| L(s) = 1 | − 0.237·5-s − 0.377·7-s − 0.671·11-s + 0.572·13-s − 0.197·17-s + 1.82·19-s + 1.41·23-s − 0.943·25-s + 1.38·29-s − 0.411·31-s + 0.0899·35-s − 1.69·37-s + 0.405·41-s − 0.0443·43-s + 0.0621·47-s + 0.142·49-s − 0.193·53-s + 0.159·55-s + 0.446·59-s − 1.34·61-s − 0.136·65-s + 0.513·67-s + 1.43·71-s − 1.06·73-s + 0.253·77-s − 0.839·79-s − 1.92·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.760161444\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.760161444\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + 0.532T + 5T^{2} \) |
| 11 | \( 1 + 2.22T + 11T^{2} \) |
| 13 | \( 1 - 2.06T + 13T^{2} \) |
| 17 | \( 1 + 0.815T + 17T^{2} \) |
| 19 | \( 1 - 7.94T + 19T^{2} \) |
| 23 | \( 1 - 6.80T + 23T^{2} \) |
| 29 | \( 1 - 7.47T + 29T^{2} \) |
| 31 | \( 1 + 2.29T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 - 2.59T + 41T^{2} \) |
| 43 | \( 1 + 0.290T + 43T^{2} \) |
| 47 | \( 1 - 0.426T + 47T^{2} \) |
| 53 | \( 1 + 1.41T + 53T^{2} \) |
| 59 | \( 1 - 3.43T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 - 4.20T + 67T^{2} \) |
| 71 | \( 1 - 12.0T + 71T^{2} \) |
| 73 | \( 1 + 9.09T + 73T^{2} \) |
| 79 | \( 1 + 7.46T + 79T^{2} \) |
| 83 | \( 1 + 17.5T + 83T^{2} \) |
| 89 | \( 1 - 2.09T + 89T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
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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
−7.61560058896578302344334582407, −7.14714694552327799355863953811, −6.42428192542730361503269565276, −5.54581193989269308545396616500, −5.10401249935893826809905281050, −4.20041969951546270606649997050, −3.27068266268950213905337288966, −2.89644965119238528042385376950, −1.65389287250209917126837319312, −0.65413206049932382237357048379,
0.65413206049932382237357048379, 1.65389287250209917126837319312, 2.89644965119238528042385376950, 3.27068266268950213905337288966, 4.20041969951546270606649997050, 5.10401249935893826809905281050, 5.54581193989269308545396616500, 6.42428192542730361503269565276, 7.14714694552327799355863953811, 7.61560058896578302344334582407
