L(s) = 1 | − 0.618·5-s + 2.23·7-s − 4.23·13-s − 0.381·17-s + 1.38·19-s + 5.47·23-s − 4.61·25-s + 2·29-s − 8.09·31-s − 1.38·35-s + 6.70·37-s + 3.76·41-s − 5.94·43-s − 9.61·47-s − 1.99·49-s + 8.56·53-s − 3.85·59-s + 6.61·61-s + 2.61·65-s − 2.90·67-s + 2.14·71-s − 8.76·73-s + 6.23·79-s − 13.1·83-s + 0.236·85-s + 7.76·89-s − 9.47·91-s + ⋯ |
L(s) = 1 | − 0.276·5-s + 0.845·7-s − 1.17·13-s − 0.0926·17-s + 0.317·19-s + 1.14·23-s − 0.923·25-s + 0.371·29-s − 1.45·31-s − 0.233·35-s + 1.10·37-s + 0.587·41-s − 0.906·43-s − 1.40·47-s − 0.285·49-s + 1.17·53-s − 0.501·59-s + 0.847·61-s + 0.324·65-s − 0.355·67-s + 0.254·71-s − 1.02·73-s + 0.701·79-s − 1.44·83-s + 0.0256·85-s + 0.822·89-s − 0.992·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
good | 5 | \( 1 + 0.618T + 5T^{2} \) |
| 7 | \( 1 - 2.23T + 7T^{2} \) |
| 13 | \( 1 + 4.23T + 13T^{2} \) |
| 17 | \( 1 + 0.381T + 17T^{2} \) |
| 19 | \( 1 - 1.38T + 19T^{2} \) |
| 23 | \( 1 - 5.47T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 8.09T + 31T^{2} \) |
| 37 | \( 1 - 6.70T + 37T^{2} \) |
| 41 | \( 1 - 3.76T + 41T^{2} \) |
| 43 | \( 1 + 5.94T + 43T^{2} \) |
| 47 | \( 1 + 9.61T + 47T^{2} \) |
| 53 | \( 1 - 8.56T + 53T^{2} \) |
| 59 | \( 1 + 3.85T + 59T^{2} \) |
| 61 | \( 1 - 6.61T + 61T^{2} \) |
| 67 | \( 1 + 2.90T + 67T^{2} \) |
| 71 | \( 1 - 2.14T + 71T^{2} \) |
| 73 | \( 1 + 8.76T + 73T^{2} \) |
| 79 | \( 1 - 6.23T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 - 7.76T + 89T^{2} \) |
| 97 | \( 1 + 17.0T + 97T^{2} \) |
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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
−7.37540752907799417102087962684, −6.98911493911391647168159195850, −5.96322822270658315484681821239, −5.20359989643507332119969118714, −4.72415761624446654676417897570, −3.94268910428250299056081628761, −3.02166783241431784562539168941, −2.19622443956242327761019287057, −1.28808931368260421699141916196, 0,
1.28808931368260421699141916196, 2.19622443956242327761019287057, 3.02166783241431784562539168941, 3.94268910428250299056081628761, 4.72415761624446654676417897570, 5.20359989643507332119969118714, 5.96322822270658315484681821239, 6.98911493911391647168159195850, 7.37540752907799417102087962684