| L(s) = 1 | − 0.141·2-s − 3-s − 1.97·4-s + 0.141·6-s − 0.858·7-s + 0.563·8-s + 9-s − 3.67·11-s + 1.97·12-s + 4.58·13-s + 0.121·14-s + 3.87·16-s − 5.30·17-s − 0.141·18-s + 6.36·19-s + 0.858·21-s + 0.521·22-s + 3.42·23-s − 0.563·24-s − 0.649·26-s − 27-s + 1.69·28-s + 3.73·29-s + 1.25·31-s − 1.67·32-s + 3.67·33-s + 0.751·34-s + ⋯ |
| L(s) = 1 | − 0.100·2-s − 0.577·3-s − 0.989·4-s + 0.0578·6-s − 0.324·7-s + 0.199·8-s + 0.333·9-s − 1.10·11-s + 0.571·12-s + 1.27·13-s + 0.0325·14-s + 0.969·16-s − 1.28·17-s − 0.0333·18-s + 1.46·19-s + 0.187·21-s + 0.111·22-s + 0.713·23-s − 0.115·24-s − 0.127·26-s − 0.192·27-s + 0.321·28-s + 0.693·29-s + 0.225·31-s − 0.296·32-s + 0.640·33-s + 0.128·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 2 | \( 1 + 0.141T + 2T^{2} \) |
| 7 | \( 1 + 0.858T + 7T^{2} \) |
| 11 | \( 1 + 3.67T + 11T^{2} \) |
| 13 | \( 1 - 4.58T + 13T^{2} \) |
| 17 | \( 1 + 5.30T + 17T^{2} \) |
| 19 | \( 1 - 6.36T + 19T^{2} \) |
| 23 | \( 1 - 3.42T + 23T^{2} \) |
| 29 | \( 1 - 3.73T + 29T^{2} \) |
| 31 | \( 1 - 1.25T + 31T^{2} \) |
| 37 | \( 1 + 7.45T + 37T^{2} \) |
| 41 | \( 1 + 2.53T + 41T^{2} \) |
| 43 | \( 1 + 3.37T + 43T^{2} \) |
| 47 | \( 1 + 8.49T + 47T^{2} \) |
| 53 | \( 1 + 2.34T + 53T^{2} \) |
| 59 | \( 1 - 13.1T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 - 3.34T + 67T^{2} \) |
| 71 | \( 1 + 4.32T + 71T^{2} \) |
| 73 | \( 1 + 9.08T + 73T^{2} \) |
| 79 | \( 1 + 3.24T + 79T^{2} \) |
| 83 | \( 1 + 7.39T + 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−8.700122947456562179508730668276, −8.317745807941576936681618086942, −7.21011543788252383893912642551, −6.42393307733547270414948656143, −5.38430546270427350274679481011, −4.95541045982864586404747110251, −3.88639494298388764539754071963, −2.97530393123205402560298374278, −1.29757094918371103392098490418, 0,
1.29757094918371103392098490418, 2.97530393123205402560298374278, 3.88639494298388764539754071963, 4.95541045982864586404747110251, 5.38430546270427350274679481011, 6.42393307733547270414948656143, 7.21011543788252383893912642551, 8.317745807941576936681618086942, 8.700122947456562179508730668276
