| L(s) = 1 | − 1.24·2-s − 3-s − 0.453·4-s + 3.27·5-s + 1.24·6-s − 7-s + 3.05·8-s + 9-s − 4.07·10-s + 5.72·11-s + 0.453·12-s − 2.61·13-s + 1.24·14-s − 3.27·15-s − 2.88·16-s + 0.475·17-s − 1.24·18-s + 3.43·19-s − 1.48·20-s + 21-s − 7.12·22-s − 6.63·23-s − 3.05·24-s + 5.72·25-s + 3.25·26-s − 27-s + 0.453·28-s + ⋯ |
| L(s) = 1 | − 0.879·2-s − 0.577·3-s − 0.226·4-s + 1.46·5-s + 0.507·6-s − 0.377·7-s + 1.07·8-s + 0.333·9-s − 1.28·10-s + 1.72·11-s + 0.131·12-s − 0.726·13-s + 0.332·14-s − 0.845·15-s − 0.721·16-s + 0.115·17-s − 0.293·18-s + 0.788·19-s − 0.332·20-s + 0.218·21-s − 1.51·22-s − 1.38·23-s − 0.622·24-s + 1.14·25-s + 0.638·26-s − 0.192·27-s + 0.0857·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9861336248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9861336248\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + 1.24T + 2T^{2} \) |
| 5 | \( 1 - 3.27T + 5T^{2} \) |
| 11 | \( 1 - 5.72T + 11T^{2} \) |
| 13 | \( 1 + 2.61T + 13T^{2} \) |
| 17 | \( 1 - 0.475T + 17T^{2} \) |
| 19 | \( 1 - 3.43T + 19T^{2} \) |
| 23 | \( 1 + 6.63T + 23T^{2} \) |
| 29 | \( 1 + 0.697T + 29T^{2} \) |
| 31 | \( 1 - 7.47T + 31T^{2} \) |
| 37 | \( 1 + 8.09T + 37T^{2} \) |
| 43 | \( 1 - 12.0T + 43T^{2} \) |
| 47 | \( 1 + 2.85T + 47T^{2} \) |
| 53 | \( 1 - 8.56T + 53T^{2} \) |
| 59 | \( 1 - 7.83T + 59T^{2} \) |
| 61 | \( 1 + 8.88T + 61T^{2} \) |
| 67 | \( 1 + 0.164T + 67T^{2} \) |
| 71 | \( 1 + 3.16T + 71T^{2} \) |
| 73 | \( 1 - 1.18T + 73T^{2} \) |
| 79 | \( 1 - 17.3T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−9.960676490309535572398117523570, −9.452451368112190252163232118364, −8.884086377419741956243615617573, −7.62744450809660657895415598570, −6.65929225491269001399651534711, −5.97169905688819869502741991418, −4.99041365637323591207381497705, −3.88875454479782303938212653293, −2.09691020579345527268710914547, −1.00787952138824009643324864320,
1.00787952138824009643324864320, 2.09691020579345527268710914547, 3.88875454479782303938212653293, 4.99041365637323591207381497705, 5.97169905688819869502741991418, 6.65929225491269001399651534711, 7.62744450809660657895415598570, 8.884086377419741956243615617573, 9.452451368112190252163232118364, 9.960676490309535572398117523570
