| L(s) = 1 | + 2.73·2-s + 5.46·4-s − 5-s − 3·7-s + 9.46·8-s − 2.73·10-s + 2.73·11-s + 6.19·13-s − 8.19·14-s + 14.9·16-s + 2.26·17-s − 3.26·19-s − 5.46·20-s + 7.46·22-s − 23-s + 25-s + 16.9·26-s − 16.3·28-s + 3.73·29-s − 9.92·31-s + 21.8·32-s + 6.19·34-s + 3·35-s − 1.92·37-s − 8.92·38-s − 9.46·40-s + 3.73·41-s + ⋯ |
| L(s) = 1 | + 1.93·2-s + 2.73·4-s − 0.447·5-s − 1.13·7-s + 3.34·8-s − 0.863·10-s + 0.823·11-s + 1.71·13-s − 2.19·14-s + 3.73·16-s + 0.550·17-s − 0.749·19-s − 1.22·20-s + 1.59·22-s − 0.208·23-s + 0.200·25-s + 3.31·26-s − 3.09·28-s + 0.693·29-s − 1.78·31-s + 3.86·32-s + 1.06·34-s + 0.507·35-s − 0.316·37-s − 1.44·38-s − 1.49·40-s + 0.582·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.937733995\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.937733995\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 2.73T + 2T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 - 2.73T + 11T^{2} \) |
| 13 | \( 1 - 6.19T + 13T^{2} \) |
| 17 | \( 1 - 2.26T + 17T^{2} \) |
| 19 | \( 1 + 3.26T + 19T^{2} \) |
| 29 | \( 1 - 3.73T + 29T^{2} \) |
| 31 | \( 1 + 9.92T + 31T^{2} \) |
| 37 | \( 1 + 1.92T + 37T^{2} \) |
| 41 | \( 1 - 3.73T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 + 7.66T + 47T^{2} \) |
| 53 | \( 1 - 5.73T + 53T^{2} \) |
| 59 | \( 1 + 7.19T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 + 6.46T + 67T^{2} \) |
| 71 | \( 1 - 3.73T + 71T^{2} \) |
| 73 | \( 1 + 7.66T + 73T^{2} \) |
| 79 | \( 1 - 6.92T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 + 2.53T + 89T^{2} \) |
| 97 | \( 1 + 4T + 97T^{2} \) |
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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
−10.36524821864218101933586939458, −9.100632435840797047544063532217, −7.988349823213483503059164425893, −6.85480805001265726967461712305, −6.36383912804470012744726917480, −5.69625395293239276746969754286, −4.48266561916194168317788589444, −3.58966053639723914347409430607, −3.26063237304381924662892604655, −1.64547586178015891322970375742,
1.64547586178015891322970375742, 3.26063237304381924662892604655, 3.58966053639723914347409430607, 4.48266561916194168317788589444, 5.69625395293239276746969754286, 6.36383912804470012744726917480, 6.85480805001265726967461712305, 7.988349823213483503059164425893, 9.100632435840797047544063532217, 10.36524821864218101933586939458
