| L(s) = 1 | − 2.74·2-s + 3-s + 5.55·4-s − 2.74·6-s − 9.78·8-s + 9-s − 4.44·11-s + 5.55·12-s − 3.61·13-s + 15.7·16-s + 4.94·17-s − 2.74·18-s + 2.74·19-s + 12.2·22-s + 4.36·23-s − 9.78·24-s + 9.94·26-s + 27-s + 0.660·29-s + 1.25·31-s − 23.7·32-s − 4.44·33-s − 13.6·34-s + 5.55·36-s + 2.16·37-s − 7.54·38-s − 3.61·39-s + ⋯ |
| L(s) = 1 | − 1.94·2-s + 0.577·3-s + 2.77·4-s − 1.12·6-s − 3.45·8-s + 0.333·9-s − 1.34·11-s + 1.60·12-s − 1.00·13-s + 3.94·16-s + 1.19·17-s − 0.647·18-s + 0.629·19-s + 2.60·22-s + 0.909·23-s − 1.99·24-s + 1.95·26-s + 0.192·27-s + 0.122·29-s + 0.225·31-s − 4.20·32-s − 0.773·33-s − 2.33·34-s + 0.926·36-s + 0.356·37-s − 1.22·38-s − 0.579·39-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
| \(L(1)\) |
\(\approx\) |
\(0.7941336339\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7941336339\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.74T + 2T^{2} \) |
| 11 | \( 1 + 4.44T + 11T^{2} \) |
| 13 | \( 1 + 3.61T + 13T^{2} \) |
| 17 | \( 1 - 4.94T + 17T^{2} \) |
| 19 | \( 1 - 2.74T + 19T^{2} \) |
| 23 | \( 1 - 4.36T + 23T^{2} \) |
| 29 | \( 1 - 0.660T + 29T^{2} \) |
| 31 | \( 1 - 1.25T + 31T^{2} \) |
| 37 | \( 1 - 2.16T + 37T^{2} \) |
| 41 | \( 1 + 5.77T + 41T^{2} \) |
| 43 | \( 1 - 9.11T + 43T^{2} \) |
| 47 | \( 1 + 6.48T + 47T^{2} \) |
| 53 | \( 1 + 7.03T + 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 - 9.41T + 61T^{2} \) |
| 67 | \( 1 + 3.06T + 67T^{2} \) |
| 71 | \( 1 - 0.277T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 + 5.53T + 79T^{2} \) |
| 83 | \( 1 - 7.17T + 83T^{2} \) |
| 89 | \( 1 + 0.828T + 89T^{2} \) |
| 97 | \( 1 + 6.20T + 97T^{2} \) |
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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
−8.429914646007503510763344577027, −7.87850899140005062871638033455, −7.49029860630159920075233918668, −6.80826546507404103238395304726, −5.75934230589223449562089406223, −4.93550991763057356628011447392, −3.19983326125179326091952477935, −2.76967664167614973146724373250, −1.79955735601967689305456687735, −0.66566166254903819218765063437,
0.66566166254903819218765063437, 1.79955735601967689305456687735, 2.76967664167614973146724373250, 3.19983326125179326091952477935, 4.93550991763057356628011447392, 5.75934230589223449562089406223, 6.80826546507404103238395304726, 7.49029860630159920075233918668, 7.87850899140005062871638033455, 8.429914646007503510763344577027
