L(s) = 1 | + 3·3-s − 1.44·5-s + 9·9-s + 46.1·11-s − 32.2·13-s − 4.33·15-s + 77.7·17-s + 12.6·19-s − 100.·23-s − 122.·25-s + 27·27-s + 213.·29-s + 42.0·31-s + 138.·33-s + 310.·37-s − 96.6·39-s + 44.0·41-s + 381.·43-s − 13.0·45-s − 358.·47-s + 233.·51-s + 184.·53-s − 66.7·55-s + 37.9·57-s + 454.·59-s + 11.8·61-s + 46.6·65-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.129·5-s + 0.333·9-s + 1.26·11-s − 0.687·13-s − 0.0746·15-s + 1.10·17-s + 0.152·19-s − 0.914·23-s − 0.983·25-s + 0.192·27-s + 1.36·29-s + 0.243·31-s + 0.729·33-s + 1.37·37-s − 0.397·39-s + 0.167·41-s + 1.35·43-s − 0.0431·45-s − 1.11·47-s + 0.640·51-s + 0.479·53-s − 0.163·55-s + 0.0882·57-s + 1.00·59-s + 0.0248·61-s + 0.0889·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.626865615\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.626865615\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 1.44T + 125T^{2} \) |
| 11 | \( 1 - 46.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 32.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 77.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 12.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 100.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 213.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 42.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 310.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 44.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 381.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 358.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 184.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 454.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 11.8T + 2.26e5T^{2} \) |
| 67 | \( 1 - 590.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 494.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 975.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 299.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.40e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 695.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 481.T + 9.12e5T^{2} \) |
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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
−9.860396495451413029185168266246, −9.672644203229253540261957427796, −8.435716917614640016286754920283, −7.75786036666844855193264594627, −6.76846647949485184499345955135, −5.78069939949090106550768378441, −4.45403947814252433894151747473, −3.60131905047754698392058913606, −2.36576510199126841404761466631, −0.994625576613004797376751722141,
0.994625576613004797376751722141, 2.36576510199126841404761466631, 3.60131905047754698392058913606, 4.45403947814252433894151747473, 5.78069939949090106550768378441, 6.76846647949485184499345955135, 7.75786036666844855193264594627, 8.435716917614640016286754920283, 9.672644203229253540261957427796, 9.860396495451413029185168266246