| L(s) = 1 | + 3-s − 2·4-s + 5-s − 3.77·7-s + 9-s + 4.77·11-s − 2·12-s + 3.77·13-s + 15-s + 4·16-s + 6·17-s + 2.77·19-s − 2·20-s − 3.77·21-s + 25-s + 27-s + 7.54·28-s + 6·29-s − 2.77·31-s + 4.77·33-s − 3.77·35-s − 2·36-s − 9.77·37-s + 3.77·39-s − 1.22·41-s + 5.77·43-s − 9.54·44-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 0.447·5-s − 1.42·7-s + 0.333·9-s + 1.43·11-s − 0.577·12-s + 1.04·13-s + 0.258·15-s + 16-s + 1.45·17-s + 0.635·19-s − 0.447·20-s − 0.823·21-s + 0.200·25-s + 0.192·27-s + 1.42·28-s + 1.11·29-s − 0.497·31-s + 0.830·33-s − 0.637·35-s − 0.333·36-s − 1.60·37-s + 0.604·39-s − 0.191·41-s + 0.880·43-s − 1.43·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.344404608\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.344404608\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 7 | \( 1 + 3.77T + 7T^{2} \) |
| 11 | \( 1 - 4.77T + 11T^{2} \) |
| 13 | \( 1 - 3.77T + 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 2.77T + 19T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 2.77T + 31T^{2} \) |
| 37 | \( 1 + 9.77T + 37T^{2} \) |
| 41 | \( 1 + 1.22T + 41T^{2} \) |
| 43 | \( 1 - 5.77T + 43T^{2} \) |
| 47 | \( 1 + 9.54T + 47T^{2} \) |
| 53 | \( 1 - 3.54T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 8.54T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 - 1.22T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 5.22T + 79T^{2} \) |
| 83 | \( 1 + 3.54T + 83T^{2} \) |
| 89 | \( 1 - 9.54T + 89T^{2} \) |
| 97 | \( 1 - 0.455T + 97T^{2} \) |
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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
−8.025509067382997402073102892422, −7.06025776672361462671205189346, −6.43890100955517788649509018332, −5.82688279545592236257372335311, −5.06985039954611854474633698763, −3.99556097205701028151750049700, −3.48711012637385706122709688196, −3.08065688998400005341796379204, −1.56247667633522838433653471988, −0.813876241960218622955094070543,
0.813876241960218622955094070543, 1.56247667633522838433653471988, 3.08065688998400005341796379204, 3.48711012637385706122709688196, 3.99556097205701028151750049700, 5.06985039954611854474633698763, 5.82688279545592236257372335311, 6.43890100955517788649509018332, 7.06025776672361462671205189346, 8.025509067382997402073102892422
