| L(s) = 1 | + 3.16·5-s + 5.16·7-s + 4·13-s − 7.16·17-s − 1.16·19-s + 23-s + 5.00·25-s − 8.32·29-s − 6.32·31-s + 16.3·35-s − 8.32·37-s + 2·41-s + 1.16·43-s − 0.324·47-s + 19.6·49-s − 5.48·53-s + 8.32·59-s + 0.324·61-s + 12.6·65-s + 1.16·67-s − 14.6·73-s + 13.1·79-s + 4·83-s − 22.6·85-s + 9.48·89-s + 20.6·91-s − 3.67·95-s + ⋯ |
| L(s) = 1 | + 1.41·5-s + 1.95·7-s + 1.10·13-s − 1.73·17-s − 0.266·19-s + 0.208·23-s + 1.00·25-s − 1.54·29-s − 1.13·31-s + 2.75·35-s − 1.36·37-s + 0.312·41-s + 0.177·43-s − 0.0473·47-s + 2.80·49-s − 0.753·53-s + 1.08·59-s + 0.0415·61-s + 1.56·65-s + 0.141·67-s − 1.71·73-s + 1.48·79-s + 0.439·83-s − 2.45·85-s + 1.00·89-s + 2.16·91-s − 0.377·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.366683913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.366683913\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 3.16T + 5T^{2} \) |
| 7 | \( 1 - 5.16T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + 7.16T + 17T^{2} \) |
| 19 | \( 1 + 1.16T + 19T^{2} \) |
| 29 | \( 1 + 8.32T + 29T^{2} \) |
| 31 | \( 1 + 6.32T + 31T^{2} \) |
| 37 | \( 1 + 8.32T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 1.16T + 43T^{2} \) |
| 47 | \( 1 + 0.324T + 47T^{2} \) |
| 53 | \( 1 + 5.48T + 53T^{2} \) |
| 59 | \( 1 - 8.32T + 59T^{2} \) |
| 61 | \( 1 - 0.324T + 61T^{2} \) |
| 67 | \( 1 - 1.16T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 14.6T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 9.48T + 89T^{2} \) |
| 97 | \( 1 + 3.67T + 97T^{2} \) |
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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
−10.43613806945451780802420112718, −9.062606422950711948310847894300, −8.814210656265356122233352271984, −7.72743195267520578806323410302, −6.67419979706049240262632824264, −5.69509868360448891261747734117, −5.03167345791127829130952304270, −3.97367952161048494125997948309, −2.14444362300537962490348661758, −1.61119223303548843822803153001,
1.61119223303548843822803153001, 2.14444362300537962490348661758, 3.97367952161048494125997948309, 5.03167345791127829130952304270, 5.69509868360448891261747734117, 6.67419979706049240262632824264, 7.72743195267520578806323410302, 8.814210656265356122233352271984, 9.062606422950711948310847894300, 10.43613806945451780802420112718
