| L(s) = 1 | − 0.618·5-s − 7-s + 0.763·11-s + 1.23·13-s − 17-s − 8.47·19-s + 7.70·23-s − 4.61·25-s + 5.70·29-s + 6.32·31-s + 0.618·35-s + 0.472·37-s + 0.0901·41-s − 12.0·43-s + 8.47·47-s + 49-s − 10.7·53-s − 0.472·55-s − 7.32·61-s − 0.763·65-s − 13.0·67-s − 10.9·71-s + 7.14·73-s − 0.763·77-s + 2.94·79-s − 15.4·83-s + 0.618·85-s + ⋯ |
| L(s) = 1 | − 0.276·5-s − 0.377·7-s + 0.230·11-s + 0.342·13-s − 0.242·17-s − 1.94·19-s + 1.60·23-s − 0.923·25-s + 1.05·29-s + 1.13·31-s + 0.104·35-s + 0.0776·37-s + 0.0140·41-s − 1.84·43-s + 1.23·47-s + 0.142·49-s − 1.48·53-s − 0.0636·55-s − 0.938·61-s − 0.0947·65-s − 1.59·67-s − 1.29·71-s + 0.836·73-s − 0.0870·77-s + 0.331·79-s − 1.69·83-s + 0.0670·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4284 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4284 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 0.618T + 5T^{2} \) |
| 11 | \( 1 - 0.763T + 11T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 19 | \( 1 + 8.47T + 19T^{2} \) |
| 23 | \( 1 - 7.70T + 23T^{2} \) |
| 29 | \( 1 - 5.70T + 29T^{2} \) |
| 31 | \( 1 - 6.32T + 31T^{2} \) |
| 37 | \( 1 - 0.472T + 37T^{2} \) |
| 41 | \( 1 - 0.0901T + 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 - 8.47T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 7.32T + 61T^{2} \) |
| 67 | \( 1 + 13.0T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 - 7.14T + 73T^{2} \) |
| 79 | \( 1 - 2.94T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−8.161456625781903797781282308856, −7.22161491746642251775579932089, −6.47082969935363561311145006349, −6.07042041327012047461167829876, −4.83469979927733478803504324273, −4.31817944276224494322445207930, −3.35676430582467401580901416883, −2.53937532242296783826903664086, −1.36865370623512525384696058792, 0,
1.36865370623512525384696058792, 2.53937532242296783826903664086, 3.35676430582467401580901416883, 4.31817944276224494322445207930, 4.83469979927733478803504324273, 6.07042041327012047461167829876, 6.47082969935363561311145006349, 7.22161491746642251775579932089, 8.161456625781903797781282308856
