| L(s) = 1 | + 2.43·2-s − 3-s + 3.94·4-s − 2.43·6-s + 4.73·8-s + 9-s + 4.58·11-s − 3.94·12-s − 1.35·13-s + 3.64·16-s − 4.29·17-s + 2.43·18-s + 6.81·19-s + 11.1·22-s + 3.88·23-s − 4.73·24-s − 3.31·26-s − 27-s + 4·29-s + 8.10·31-s − 0.569·32-s − 4.58·33-s − 10.4·34-s + 3.94·36-s − 11.7·37-s + 16.6·38-s + 1.35·39-s + ⋯ |
| L(s) = 1 | + 1.72·2-s − 0.577·3-s + 1.97·4-s − 0.995·6-s + 1.67·8-s + 0.333·9-s + 1.38·11-s − 1.13·12-s − 0.376·13-s + 0.911·16-s − 1.04·17-s + 0.574·18-s + 1.56·19-s + 2.38·22-s + 0.809·23-s − 0.965·24-s − 0.649·26-s − 0.192·27-s + 0.742·29-s + 1.45·31-s − 0.100·32-s − 0.797·33-s − 1.79·34-s + 0.656·36-s − 1.93·37-s + 2.69·38-s + 0.217·39-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
| \(L(1)\) |
\(\approx\) |
\(5.035803771\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.035803771\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.43T + 2T^{2} \) |
| 11 | \( 1 - 4.58T + 11T^{2} \) |
| 13 | \( 1 + 1.35T + 13T^{2} \) |
| 17 | \( 1 + 4.29T + 17T^{2} \) |
| 19 | \( 1 - 6.81T + 19T^{2} \) |
| 23 | \( 1 - 3.88T + 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 - 8.10T + 31T^{2} \) |
| 37 | \( 1 + 11.7T + 37T^{2} \) |
| 41 | \( 1 + 8.17T + 41T^{2} \) |
| 43 | \( 1 - 3.52T + 43T^{2} \) |
| 47 | \( 1 - 3.29T + 47T^{2} \) |
| 53 | \( 1 - 9.45T + 53T^{2} \) |
| 59 | \( 1 + 0.700T + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 - 1.94T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 - 6.16T + 73T^{2} \) |
| 79 | \( 1 + 6.16T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 + 0.292T + 89T^{2} \) |
| 97 | \( 1 + 2.00T + 97T^{2} \) |
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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
−8.492168856990156342676613628097, −7.17361605694647162551967093184, −6.82241819941760872812187627657, −6.23223056910987132117957731919, −5.25667456811293638583315219380, −4.87099830372604334086163369089, −3.99937524080539103449705512100, −3.32400727565855169226455093682, −2.30990119237801905138515882506, −1.11819745092680352872338581209,
1.11819745092680352872338581209, 2.30990119237801905138515882506, 3.32400727565855169226455093682, 3.99937524080539103449705512100, 4.87099830372604334086163369089, 5.25667456811293638583315219380, 6.23223056910987132117957731919, 6.82241819941760872812187627657, 7.17361605694647162551967093184, 8.492168856990156342676613628097
