L(s) = 1 | − 1.30·2-s − 0.285·4-s + 5-s + 4.28·7-s + 2.99·8-s − 1.30·10-s − 3.71·11-s + 1.17·13-s − 5.60·14-s − 3.34·16-s − 7.72·17-s − 0.216·19-s − 0.285·20-s + 4.86·22-s + 1.27·23-s + 25-s − 1.54·26-s − 1.22·28-s + 5.32·29-s − 3.94·31-s − 1.60·32-s + 10.1·34-s + 4.28·35-s + 0.356·37-s + 0.283·38-s + 2.99·40-s + 11.1·41-s + ⋯ |
L(s) = 1 | − 0.925·2-s − 0.142·4-s + 0.447·5-s + 1.61·7-s + 1.05·8-s − 0.414·10-s − 1.11·11-s + 0.327·13-s − 1.49·14-s − 0.836·16-s − 1.87·17-s − 0.0496·19-s − 0.0639·20-s + 1.03·22-s + 0.266·23-s + 0.200·25-s − 0.302·26-s − 0.231·28-s + 0.989·29-s − 0.709·31-s − 0.283·32-s + 1.73·34-s + 0.723·35-s + 0.0586·37-s + 0.0459·38-s + 0.473·40-s + 1.73·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.203598675\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.203598675\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 89 | \( 1 + T \) |
good | 2 | \( 1 + 1.30T + 2T^{2} \) |
| 7 | \( 1 - 4.28T + 7T^{2} \) |
| 11 | \( 1 + 3.71T + 11T^{2} \) |
| 13 | \( 1 - 1.17T + 13T^{2} \) |
| 17 | \( 1 + 7.72T + 17T^{2} \) |
| 19 | \( 1 + 0.216T + 19T^{2} \) |
| 23 | \( 1 - 1.27T + 23T^{2} \) |
| 29 | \( 1 - 5.32T + 29T^{2} \) |
| 31 | \( 1 + 3.94T + 31T^{2} \) |
| 37 | \( 1 - 0.356T + 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 - 3.34T + 43T^{2} \) |
| 47 | \( 1 + 2.32T + 47T^{2} \) |
| 53 | \( 1 + 5.58T + 53T^{2} \) |
| 59 | \( 1 - 1.34T + 59T^{2} \) |
| 61 | \( 1 - 3.59T + 61T^{2} \) |
| 67 | \( 1 - 8.60T + 67T^{2} \) |
| 71 | \( 1 - 7.59T + 71T^{2} \) |
| 73 | \( 1 - 3.56T + 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 + 9.17T + 83T^{2} \) |
| 97 | \( 1 - 7.73T + 97T^{2} \) |
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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
−8.478802637532919847775461558216, −7.936791561208694562089923811483, −7.30341240680289764955543545514, −6.36104404347815590242927348379, −5.26023700476312286134412066557, −4.79264986346817339430276485618, −4.09246151231686625211332843477, −2.47819702425603774511785989748, −1.86059069091535710308164456738, −0.74137341731732798148969113237,
0.74137341731732798148969113237, 1.86059069091535710308164456738, 2.47819702425603774511785989748, 4.09246151231686625211332843477, 4.79264986346817339430276485618, 5.26023700476312286134412066557, 6.36104404347815590242927348379, 7.30341240680289764955543545514, 7.936791561208694562089923811483, 8.478802637532919847775461558216