| L(s) = 1 | + 1.32·2-s − 1.56·3-s − 0.231·4-s − 5-s − 2.07·6-s − 2.96·8-s − 0.561·9-s − 1.32·10-s − 0.659·11-s + 0.362·12-s − 5.91·13-s + 1.56·15-s − 3.48·16-s − 0.844·17-s − 0.746·18-s − 0.659·19-s + 0.231·20-s − 0.876·22-s − 23-s + 4.63·24-s + 25-s − 7.85·26-s + 5.56·27-s + 1.59·29-s + 2.07·30-s − 6.75·31-s + 1.30·32-s + ⋯ |
| L(s) = 1 | + 0.940·2-s − 0.901·3-s − 0.115·4-s − 0.447·5-s − 0.847·6-s − 1.04·8-s − 0.187·9-s − 0.420·10-s − 0.198·11-s + 0.104·12-s − 1.63·13-s + 0.403·15-s − 0.870·16-s − 0.204·17-s − 0.176·18-s − 0.151·19-s + 0.0518·20-s − 0.186·22-s − 0.208·23-s + 0.945·24-s + 0.200·25-s − 1.54·26-s + 1.07·27-s + 0.295·29-s + 0.379·30-s − 1.21·31-s + 0.230·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5635 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5635 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3812251424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3812251424\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 1.32T + 2T^{2} \) |
| 3 | \( 1 + 1.56T + 3T^{2} \) |
| 11 | \( 1 + 0.659T + 11T^{2} \) |
| 13 | \( 1 + 5.91T + 13T^{2} \) |
| 17 | \( 1 + 0.844T + 17T^{2} \) |
| 19 | \( 1 + 0.659T + 19T^{2} \) |
| 29 | \( 1 - 1.59T + 29T^{2} \) |
| 31 | \( 1 + 6.75T + 31T^{2} \) |
| 37 | \( 1 + 11.7T + 37T^{2} \) |
| 41 | \( 1 + 6.40T + 41T^{2} \) |
| 43 | \( 1 + 9.47T + 43T^{2} \) |
| 47 | \( 1 + 6.88T + 47T^{2} \) |
| 53 | \( 1 - 6.64T + 53T^{2} \) |
| 59 | \( 1 - 4.97T + 59T^{2} \) |
| 61 | \( 1 + 5.78T + 61T^{2} \) |
| 67 | \( 1 - 8.31T + 67T^{2} \) |
| 71 | \( 1 + 3.63T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 - 1.02T + 79T^{2} \) |
| 83 | \( 1 - 8.27T + 83T^{2} \) |
| 89 | \( 1 + 17.6T + 89T^{2} \) |
| 97 | \( 1 - 8.34T + 97T^{2} \) |
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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
−8.159311590988209692016531036016, −7.05290634399305088498371245001, −6.68201157115751028481541456616, −5.69225266859312595812989510203, −5.04896477085325708792873434457, −4.85384729502977137850064499114, −3.77809847704870861284043773721, −3.08844230924598182402032248452, −2.06091244418807457047036398120, −0.27948560662752037497574364098,
0.27948560662752037497574364098, 2.06091244418807457047036398120, 3.08844230924598182402032248452, 3.77809847704870861284043773721, 4.85384729502977137850064499114, 5.04896477085325708792873434457, 5.69225266859312595812989510203, 6.68201157115751028481541456616, 7.05290634399305088498371245001, 8.159311590988209692016531036016
