| L(s) = 1 | − 0.287·2-s + 3-s − 1.91·4-s − 0.287·6-s + 1.12·8-s + 9-s − 3.33·11-s − 1.91·12-s + 4.54·13-s + 3.51·16-s − 5.54·17-s − 0.287·18-s + 1.65·19-s + 0.956·22-s − 7.63·23-s + 1.12·24-s − 1.30·26-s + 27-s − 0.118·29-s + 6.26·31-s − 3.26·32-s − 3.33·33-s + 1.59·34-s − 1.91·36-s + 7.75·37-s − 0.476·38-s + 4.54·39-s + ⋯ |
| L(s) = 1 | − 0.203·2-s + 0.577·3-s − 0.958·4-s − 0.117·6-s + 0.397·8-s + 0.333·9-s − 1.00·11-s − 0.553·12-s + 1.26·13-s + 0.877·16-s − 1.34·17-s − 0.0677·18-s + 0.380·19-s + 0.204·22-s − 1.59·23-s + 0.229·24-s − 0.256·26-s + 0.192·27-s − 0.0220·29-s + 1.12·31-s − 0.576·32-s − 0.579·33-s + 0.273·34-s − 0.319·36-s + 1.27·37-s − 0.0772·38-s + 0.728·39-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
| \(L(1)\) |
\(\approx\) |
\(1.444780281\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.444780281\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.287T + 2T^{2} \) |
| 11 | \( 1 + 3.33T + 11T^{2} \) |
| 13 | \( 1 - 4.54T + 13T^{2} \) |
| 17 | \( 1 + 5.54T + 17T^{2} \) |
| 19 | \( 1 - 1.65T + 19T^{2} \) |
| 23 | \( 1 + 7.63T + 23T^{2} \) |
| 29 | \( 1 + 0.118T + 29T^{2} \) |
| 31 | \( 1 - 6.26T + 31T^{2} \) |
| 37 | \( 1 - 7.75T + 37T^{2} \) |
| 41 | \( 1 + 0.0701T + 41T^{2} \) |
| 43 | \( 1 - 2.92T + 43T^{2} \) |
| 47 | \( 1 - 6.38T + 47T^{2} \) |
| 53 | \( 1 - 0.739T + 53T^{2} \) |
| 59 | \( 1 - 1.63T + 59T^{2} \) |
| 61 | \( 1 - 7.31T + 61T^{2} \) |
| 67 | \( 1 - 3.03T + 67T^{2} \) |
| 71 | \( 1 + 3.77T + 71T^{2} \) |
| 73 | \( 1 + 2.35T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 + 1.22T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 - 3.04T + 97T^{2} \) |
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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
−8.384867868428256183251069292576, −8.155013665902469328615463439236, −7.28527817544257207691218130116, −6.21536203683196091427932098520, −5.55744163884573710015568280904, −4.45315122789529564647575889825, −4.06578508198054488877092597398, −3.00700869999384132679884919248, −2.01922528300175350015662260436, −0.70789191304017678213872932895,
0.70789191304017678213872932895, 2.01922528300175350015662260436, 3.00700869999384132679884919248, 4.06578508198054488877092597398, 4.45315122789529564647575889825, 5.55744163884573710015568280904, 6.21536203683196091427932098520, 7.28527817544257207691218130116, 8.155013665902469328615463439236, 8.384867868428256183251069292576
