| L(s) = 1 | − 1.34·2-s − 2.02·3-s − 0.202·4-s + 2.71·6-s − 7-s + 2.95·8-s + 1.11·9-s − 4.63·11-s + 0.411·12-s + 1.05·13-s + 1.34·14-s − 3.55·16-s − 0.442·17-s − 1.49·18-s − 0.296·19-s + 2.02·21-s + 6.21·22-s − 23-s − 5.98·24-s − 1.40·26-s + 3.82·27-s + 0.202·28-s − 6.48·29-s − 2.99·31-s − 1.14·32-s + 9.39·33-s + 0.593·34-s + ⋯ |
| L(s) = 1 | − 0.947·2-s − 1.17·3-s − 0.101·4-s + 1.10·6-s − 0.377·7-s + 1.04·8-s + 0.370·9-s − 1.39·11-s + 0.118·12-s + 0.291·13-s + 0.358·14-s − 0.888·16-s − 0.107·17-s − 0.351·18-s − 0.0679·19-s + 0.442·21-s + 1.32·22-s − 0.208·23-s − 1.22·24-s − 0.276·26-s + 0.736·27-s + 0.0383·28-s − 1.20·29-s − 0.538·31-s − 0.202·32-s + 1.63·33-s + 0.101·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1573906078\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1573906078\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 1.34T + 2T^{2} \) |
| 3 | \( 1 + 2.02T + 3T^{2} \) |
| 11 | \( 1 + 4.63T + 11T^{2} \) |
| 13 | \( 1 - 1.05T + 13T^{2} \) |
| 17 | \( 1 + 0.442T + 17T^{2} \) |
| 19 | \( 1 + 0.296T + 19T^{2} \) |
| 29 | \( 1 + 6.48T + 29T^{2} \) |
| 31 | \( 1 + 2.99T + 31T^{2} \) |
| 37 | \( 1 + 0.971T + 37T^{2} \) |
| 41 | \( 1 + 6.58T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 + 3.92T + 47T^{2} \) |
| 53 | \( 1 + 0.592T + 53T^{2} \) |
| 59 | \( 1 - 6.10T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 - 0.569T + 71T^{2} \) |
| 73 | \( 1 + 9.38T + 73T^{2} \) |
| 79 | \( 1 + 1.57T + 79T^{2} \) |
| 83 | \( 1 + 5.03T + 83T^{2} \) |
| 89 | \( 1 + 3.61T + 89T^{2} \) |
| 97 | \( 1 + 3.01T + 97T^{2} \) |
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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
−8.522721897040852756510853298205, −7.68195595734319135502235966998, −7.18702470758725543163870895914, −6.16713869875688005799460913881, −5.52741738457141345942797035731, −4.89803893874248463809266725403, −3.99536352145091794256431868516, −2.78460181838121551937889153299, −1.55907660771002825050769338822, −0.28317807950632731954216403855,
0.28317807950632731954216403855, 1.55907660771002825050769338822, 2.78460181838121551937889153299, 3.99536352145091794256431868516, 4.89803893874248463809266725403, 5.52741738457141345942797035731, 6.16713869875688005799460913881, 7.18702470758725543163870895914, 7.68195595734319135502235966998, 8.522721897040852756510853298205
