| L(s) = 1 | − 1.55·2-s − 3-s + 0.409·4-s + 1.55·6-s + 2.46·8-s + 9-s + 4.43·11-s − 0.409·12-s − 1.73·13-s − 4.65·16-s + 2.73·17-s − 1.55·18-s + 0.305·19-s − 6.87·22-s − 7.02·23-s − 2.46·24-s + 2.68·26-s − 27-s − 7.79·29-s + 5.28·31-s + 2.28·32-s − 4.43·33-s − 4.24·34-s + 0.409·36-s − 3.67·37-s − 0.473·38-s + 1.73·39-s + ⋯ |
| L(s) = 1 | − 1.09·2-s − 0.577·3-s + 0.204·4-s + 0.633·6-s + 0.872·8-s + 0.333·9-s + 1.33·11-s − 0.118·12-s − 0.480·13-s − 1.16·16-s + 0.662·17-s − 0.365·18-s + 0.0700·19-s − 1.46·22-s − 1.46·23-s − 0.503·24-s + 0.527·26-s − 0.192·27-s − 1.44·29-s + 0.949·31-s + 0.403·32-s − 0.771·33-s − 0.727·34-s + 0.0683·36-s − 0.604·37-s − 0.0768·38-s + 0.277·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.55T + 2T^{2} \) |
| 11 | \( 1 - 4.43T + 11T^{2} \) |
| 13 | \( 1 + 1.73T + 13T^{2} \) |
| 17 | \( 1 - 2.73T + 17T^{2} \) |
| 19 | \( 1 - 0.305T + 19T^{2} \) |
| 23 | \( 1 + 7.02T + 23T^{2} \) |
| 29 | \( 1 + 7.79T + 29T^{2} \) |
| 31 | \( 1 - 5.28T + 31T^{2} \) |
| 37 | \( 1 + 3.67T + 37T^{2} \) |
| 41 | \( 1 - 6.71T + 41T^{2} \) |
| 43 | \( 1 + 9.71T + 43T^{2} \) |
| 47 | \( 1 + 1.81T + 47T^{2} \) |
| 53 | \( 1 + 1.71T + 53T^{2} \) |
| 59 | \( 1 - 1.14T + 59T^{2} \) |
| 61 | \( 1 + 9.55T + 61T^{2} \) |
| 67 | \( 1 + 8.38T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 79 | \( 1 - 6.71T + 79T^{2} \) |
| 83 | \( 1 + 5.09T + 83T^{2} \) |
| 89 | \( 1 + 4.07T + 89T^{2} \) |
| 97 | \( 1 - 2.87T + 97T^{2} \) |
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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
−8.136116678843531677797750676832, −7.61185149322554404005787850604, −6.78942922984331719392784529353, −6.11367858309634248337483490673, −5.17756905325342744477378603752, −4.31549673984845122112757752333, −3.57813695908242907080431666329, −2.02148080762505461599550543113, −1.19918220318968613516065468153, 0,
1.19918220318968613516065468153, 2.02148080762505461599550543113, 3.57813695908242907080431666329, 4.31549673984845122112757752333, 5.17756905325342744477378603752, 6.11367858309634248337483490673, 6.78942922984331719392784529353, 7.61185149322554404005787850604, 8.136116678843531677797750676832
