| L(s) = 1 | − 2-s + 4-s + 3.70·5-s + 7-s − 8-s − 3.70·10-s − 5.70·11-s + 13-s − 14-s + 16-s − 3.70·17-s + 5.70·19-s + 3.70·20-s + 5.70·22-s + 1.70·23-s + 8.70·25-s − 26-s + 28-s + 3.70·29-s − 32-s + 3.70·34-s + 3.70·35-s + 4.29·37-s − 5.70·38-s − 3.70·40-s + 9.40·41-s + 9.10·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.65·5-s + 0.377·7-s − 0.353·8-s − 1.17·10-s − 1.71·11-s + 0.277·13-s − 0.267·14-s + 0.250·16-s − 0.897·17-s + 1.30·19-s + 0.827·20-s + 1.21·22-s + 0.354·23-s + 1.74·25-s − 0.196·26-s + 0.188·28-s + 0.687·29-s − 0.176·32-s + 0.634·34-s + 0.625·35-s + 0.706·37-s − 0.924·38-s − 0.585·40-s + 1.46·41-s + 1.38·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.680378657\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.680378657\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 3.70T + 5T^{2} \) |
| 11 | \( 1 + 5.70T + 11T^{2} \) |
| 17 | \( 1 + 3.70T + 17T^{2} \) |
| 19 | \( 1 - 5.70T + 19T^{2} \) |
| 23 | \( 1 - 1.70T + 23T^{2} \) |
| 29 | \( 1 - 3.70T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 4.29T + 37T^{2} \) |
| 41 | \( 1 - 9.40T + 41T^{2} \) |
| 43 | \( 1 - 9.10T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 - 7.70T + 61T^{2} \) |
| 67 | \( 1 + 7.40T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 7.70T + 73T^{2} \) |
| 79 | \( 1 + 3.40T + 79T^{2} \) |
| 83 | \( 1 - 0.596T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 - 16.8T + 97T^{2} \) |
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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
−9.390783672766022918018894661632, −8.779979858745485724139932563583, −7.81378514019804717931066686645, −7.13837934149295601917379291274, −6.00227292216705132091218800281, −5.55716856392490862171145576983, −4.62191857179989800892526889459, −2.81400997151887775197998714912, −2.30095244489171223406552275491, −1.04596359762318593978200431068,
1.04596359762318593978200431068, 2.30095244489171223406552275491, 2.81400997151887775197998714912, 4.62191857179989800892526889459, 5.55716856392490862171145576983, 6.00227292216705132091218800281, 7.13837934149295601917379291274, 7.81378514019804717931066686645, 8.779979858745485724139932563583, 9.390783672766022918018894661632
