| L(s) = 1 | + 4.25·2-s − 22.3·3-s − 13.9·4-s − 94.8·6-s + 47.8·7-s − 195.·8-s + 254.·9-s + 701.·11-s + 310.·12-s + 193.·13-s + 203.·14-s − 384.·16-s + 2.30e3·17-s + 1.08e3·18-s + 2.64e3·19-s − 1.06e3·21-s + 2.98e3·22-s − 285.·23-s + 4.35e3·24-s + 824.·26-s − 253.·27-s − 666.·28-s − 5.88e3·29-s − 2.95e3·31-s + 4.61e3·32-s − 1.56e4·33-s + 9.79e3·34-s + ⋯ |
| L(s) = 1 | + 0.751·2-s − 1.43·3-s − 0.435·4-s − 1.07·6-s + 0.369·7-s − 1.07·8-s + 1.04·9-s + 1.74·11-s + 0.622·12-s + 0.318·13-s + 0.277·14-s − 0.375·16-s + 1.93·17-s + 0.786·18-s + 1.67·19-s − 0.527·21-s + 1.31·22-s − 0.112·23-s + 1.54·24-s + 0.239·26-s − 0.0670·27-s − 0.160·28-s − 1.29·29-s − 0.552·31-s + 0.796·32-s − 2.49·33-s + 1.45·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.165019724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.165019724\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 43 | \( 1 + 1.84e3T \) |
| good | 2 | \( 1 - 4.25T + 32T^{2} \) |
| 3 | \( 1 + 22.3T + 243T^{2} \) |
| 7 | \( 1 - 47.8T + 1.68e4T^{2} \) |
| 11 | \( 1 - 701.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 193.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.30e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.64e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 285.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.88e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.95e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.08e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.25e4T + 1.15e8T^{2} \) |
| 47 | \( 1 + 2.58e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.80e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.97e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.91e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.94e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.05e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.80e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.95e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.09e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.27e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 7.32e4T + 8.58e9T^{2} \) |
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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
−9.452900341167293160888619412015, −8.301202273758764807544461712024, −7.18092971757353424875218813843, −6.31158155962827522697607539797, −5.44365403203359077885691291379, −5.21999531643201433053419635008, −3.94925313120547162170799328604, −3.41089940419702697471530222844, −1.38151378696327976652980334948, −0.70428867126473387351463981481,
0.70428867126473387351463981481, 1.38151378696327976652980334948, 3.41089940419702697471530222844, 3.94925313120547162170799328604, 5.21999531643201433053419635008, 5.44365403203359077885691291379, 6.31158155962827522697607539797, 7.18092971757353424875218813843, 8.301202273758764807544461712024, 9.452900341167293160888619412015
