| L(s) = 1 | + 0.445·2-s + 3-s − 1.80·4-s − 0.246·5-s + 0.445·6-s + 1.75·7-s − 1.69·8-s + 9-s − 0.109·10-s + 5.65·11-s − 1.80·12-s + 0.780·14-s − 0.246·15-s + 2.85·16-s − 3.80·17-s + 0.445·18-s + 5.58·19-s + 0.445·20-s + 1.75·21-s + 2.51·22-s + 8.34·23-s − 1.69·24-s − 4.93·25-s + 27-s − 3.15·28-s − 5.93·29-s − 0.109·30-s + ⋯ |
| L(s) = 1 | + 0.314·2-s + 0.577·3-s − 0.900·4-s − 0.110·5-s + 0.181·6-s + 0.662·7-s − 0.598·8-s + 0.333·9-s − 0.0347·10-s + 1.70·11-s − 0.520·12-s + 0.208·14-s − 0.0637·15-s + 0.712·16-s − 0.922·17-s + 0.104·18-s + 1.28·19-s + 0.0995·20-s + 0.382·21-s + 0.536·22-s + 1.74·23-s − 0.345·24-s − 0.987·25-s + 0.192·27-s − 0.596·28-s − 1.10·29-s − 0.0200·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.851015433\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.851015433\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.445T + 2T^{2} \) |
| 5 | \( 1 + 0.246T + 5T^{2} \) |
| 7 | \( 1 - 1.75T + 7T^{2} \) |
| 11 | \( 1 - 5.65T + 11T^{2} \) |
| 17 | \( 1 + 3.80T + 17T^{2} \) |
| 19 | \( 1 - 5.58T + 19T^{2} \) |
| 23 | \( 1 - 8.34T + 23T^{2} \) |
| 29 | \( 1 + 5.93T + 29T^{2} \) |
| 31 | \( 1 - 5.26T + 31T^{2} \) |
| 37 | \( 1 - 3.19T + 37T^{2} \) |
| 41 | \( 1 - 0.445T + 41T^{2} \) |
| 43 | \( 1 - 1.71T + 43T^{2} \) |
| 47 | \( 1 + 6.73T + 47T^{2} \) |
| 53 | \( 1 + 1.06T + 53T^{2} \) |
| 59 | \( 1 + 13.7T + 59T^{2} \) |
| 61 | \( 1 + 8.51T + 61T^{2} \) |
| 67 | \( 1 - 5.96T + 67T^{2} \) |
| 71 | \( 1 + 5.71T + 71T^{2} \) |
| 73 | \( 1 - 7.35T + 73T^{2} \) |
| 79 | \( 1 - 4.45T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 - 0.137T + 89T^{2} \) |
| 97 | \( 1 + 13.6T + 97T^{2} \) |
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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
−11.12002966754360133829965115475, −9.561021900356378255153181199683, −9.262435606503044850961956068134, −8.366793768393057078528770929787, −7.39260193239174232205051567090, −6.25426652103977034272802957201, −4.97181311706309888084725703013, −4.17826828325452043239009646316, −3.20298903581918346857264570917, −1.36618448393371556817901132706,
1.36618448393371556817901132706, 3.20298903581918346857264570917, 4.17826828325452043239009646316, 4.97181311706309888084725703013, 6.25426652103977034272802957201, 7.39260193239174232205051567090, 8.366793768393057078528770929787, 9.262435606503044850961956068134, 9.561021900356378255153181199683, 11.12002966754360133829965115475
