| L(s) = 1 | + 1.61·3-s − 5-s + 0.618·7-s − 0.381·9-s − 2.85·11-s + 7.09·13-s − 1.61·15-s + 6.09·17-s + 1.85·19-s + 1.00·21-s − 23-s + 25-s − 5.47·27-s + 9.23·29-s − 9.09·31-s − 4.61·33-s − 0.618·35-s − 6.47·37-s + 11.4·39-s + 3.32·41-s + 0.381·45-s + 3.70·47-s − 6.61·49-s + 9.85·51-s − 0.472·53-s + 2.85·55-s + 3·57-s + ⋯ |
| L(s) = 1 | + 0.934·3-s − 0.447·5-s + 0.233·7-s − 0.127·9-s − 0.860·11-s + 1.96·13-s − 0.417·15-s + 1.47·17-s + 0.425·19-s + 0.218·21-s − 0.208·23-s + 0.200·25-s − 1.05·27-s + 1.71·29-s − 1.63·31-s − 0.803·33-s − 0.104·35-s − 1.06·37-s + 1.83·39-s + 0.519·41-s + 0.0569·45-s + 0.540·47-s − 0.945·49-s + 1.37·51-s − 0.0648·53-s + 0.384·55-s + 0.397·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.864449314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.864449314\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 1.61T + 3T^{2} \) |
| 7 | \( 1 - 0.618T + 7T^{2} \) |
| 11 | \( 1 + 2.85T + 11T^{2} \) |
| 13 | \( 1 - 7.09T + 13T^{2} \) |
| 17 | \( 1 - 6.09T + 17T^{2} \) |
| 19 | \( 1 - 1.85T + 19T^{2} \) |
| 29 | \( 1 - 9.23T + 29T^{2} \) |
| 31 | \( 1 + 9.09T + 31T^{2} \) |
| 37 | \( 1 + 6.47T + 37T^{2} \) |
| 41 | \( 1 - 3.32T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 3.70T + 47T^{2} \) |
| 53 | \( 1 + 0.472T + 53T^{2} \) |
| 59 | \( 1 - 1.70T + 59T^{2} \) |
| 61 | \( 1 - 9.32T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 - 4.09T + 71T^{2} \) |
| 73 | \( 1 - 3.23T + 73T^{2} \) |
| 79 | \( 1 + 1.52T + 79T^{2} \) |
| 83 | \( 1 + 6.94T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 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
−8.092545752927417875603902820627, −7.47262979578302811213487256580, −6.59111876934830152893849690710, −5.65729697233838068584139532940, −5.23925111121201693835090570713, −4.01565402326033819922029811916, −3.46183733204483374424908397305, −2.92938579063873202036819007700, −1.84583387825285221848584750045, −0.837588161432488574259431249259,
0.837588161432488574259431249259, 1.84583387825285221848584750045, 2.92938579063873202036819007700, 3.46183733204483374424908397305, 4.01565402326033819922029811916, 5.23925111121201693835090570713, 5.65729697233838068584139532940, 6.59111876934830152893849690710, 7.47262979578302811213487256580, 8.092545752927417875603902820627
