| 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
| \(L(1)\) |
\(\approx\) |
\(7.445626784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.445626784\) |
| \(L(\frac{3}{2})\) |
|
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
−8.625040754967058889256528271952, −7.43802649177841750236264759999, −6.85124325275500938475425554487, −6.18286467821691173842926500918, −5.45592868743314848570689061454, −4.53743068132376403411870763756, −3.84748256780947505613342800496, −3.36954725122701693280640375883, −2.33966052127041127078406232025, −1.40158335348539744536863209348,
1.40158335348539744536863209348, 2.33966052127041127078406232025, 3.36954725122701693280640375883, 3.84748256780947505613342800496, 4.53743068132376403411870763756, 5.45592868743314848570689061454, 6.18286467821691173842926500918, 6.85124325275500938475425554487, 7.43802649177841750236264759999, 8.625040754967058889256528271952
