| L(s) = 1 | + 2.67·2-s + 3-s + 5.15·4-s + 2.67·6-s + 2.80·7-s + 8.44·8-s + 9-s + 5.15·12-s − 5.11·13-s + 7.50·14-s + 12.2·16-s + 4.54·17-s + 2.67·18-s + 4.57·19-s + 2.80·21-s − 4·23-s + 8.44·24-s − 13.6·26-s + 27-s + 14.4·28-s + 2.38·29-s − 0.962·31-s + 15.9·32-s + 12.1·34-s + 5.15·36-s − 1.61·37-s + 12.2·38-s + ⋯ |
| L(s) = 1 | + 1.89·2-s + 0.577·3-s + 2.57·4-s + 1.09·6-s + 1.06·7-s + 2.98·8-s + 0.333·9-s + 1.48·12-s − 1.41·13-s + 2.00·14-s + 3.06·16-s + 1.10·17-s + 0.630·18-s + 1.04·19-s + 0.612·21-s − 0.834·23-s + 1.72·24-s − 2.68·26-s + 0.192·27-s + 2.73·28-s + 0.443·29-s − 0.172·31-s + 2.81·32-s + 2.08·34-s + 0.859·36-s − 0.265·37-s + 1.98·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.08359062\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.08359062\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.67T + 2T^{2} \) |
| 7 | \( 1 - 2.80T + 7T^{2} \) |
| 13 | \( 1 + 5.11T + 13T^{2} \) |
| 17 | \( 1 - 4.54T + 17T^{2} \) |
| 19 | \( 1 - 4.57T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 2.38T + 29T^{2} \) |
| 31 | \( 1 + 0.962T + 31T^{2} \) |
| 37 | \( 1 + 1.61T + 37T^{2} \) |
| 41 | \( 1 - 2.38T + 41T^{2} \) |
| 43 | \( 1 - 2.80T + 43T^{2} \) |
| 47 | \( 1 - 4.31T + 47T^{2} \) |
| 53 | \( 1 - 6.57T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 + 7.92T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 + 7.35T + 71T^{2} \) |
| 73 | \( 1 - 6.41T + 73T^{2} \) |
| 79 | \( 1 + 1.35T + 79T^{2} \) |
| 83 | \( 1 + 0.806T + 83T^{2} \) |
| 89 | \( 1 + 2.96T + 89T^{2} \) |
| 97 | \( 1 - 9.92T + 97T^{2} \) |
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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
−7.61828139325699117336046659473, −7.09984165986404169033854273957, −6.04263812970950156136323919279, −5.50010325320873716703442156154, −4.79002671290631610833740203557, −4.41398244738947921079902727578, −3.48285827296203539637603309202, −2.86039017852520544331053773583, −2.12674544853775588197824821184, −1.32750749544657828394095531713,
1.32750749544657828394095531713, 2.12674544853775588197824821184, 2.86039017852520544331053773583, 3.48285827296203539637603309202, 4.41398244738947921079902727578, 4.79002671290631610833740203557, 5.50010325320873716703442156154, 6.04263812970950156136323919279, 7.09984165986404169033854273957, 7.61828139325699117336046659473
