| L(s) = 1 | + 2.47·2-s + 4.12·4-s + 0.973·7-s + 5.25·8-s + 5.38·11-s + 1.99·13-s + 2.40·14-s + 4.74·16-s − 2.04·17-s + 6.20·19-s + 13.3·22-s + 1.93·23-s + 4.94·26-s + 4.01·28-s − 4.81·29-s − 6.64·31-s + 1.24·32-s − 5.05·34-s + 0.978·37-s + 15.3·38-s − 2.73·41-s − 3.99·43-s + 22.1·44-s + 4.79·46-s + 7.21·47-s − 6.05·49-s + 8.23·52-s + ⋯ |
| L(s) = 1 | + 1.74·2-s + 2.06·4-s + 0.367·7-s + 1.85·8-s + 1.62·11-s + 0.553·13-s + 0.643·14-s + 1.18·16-s − 0.495·17-s + 1.42·19-s + 2.83·22-s + 0.404·23-s + 0.968·26-s + 0.758·28-s − 0.894·29-s − 1.19·31-s + 0.220·32-s − 0.866·34-s + 0.160·37-s + 2.49·38-s − 0.426·41-s − 0.609·43-s + 3.34·44-s + 0.707·46-s + 1.05·47-s − 0.864·49-s + 1.14·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.338526903\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.338526903\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 2.47T + 2T^{2} \) |
| 7 | \( 1 - 0.973T + 7T^{2} \) |
| 11 | \( 1 - 5.38T + 11T^{2} \) |
| 13 | \( 1 - 1.99T + 13T^{2} \) |
| 17 | \( 1 + 2.04T + 17T^{2} \) |
| 19 | \( 1 - 6.20T + 19T^{2} \) |
| 23 | \( 1 - 1.93T + 23T^{2} \) |
| 29 | \( 1 + 4.81T + 29T^{2} \) |
| 31 | \( 1 + 6.64T + 31T^{2} \) |
| 37 | \( 1 - 0.978T + 37T^{2} \) |
| 41 | \( 1 + 2.73T + 41T^{2} \) |
| 43 | \( 1 + 3.99T + 43T^{2} \) |
| 47 | \( 1 - 7.21T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 + 6.54T + 59T^{2} \) |
| 61 | \( 1 - 2.72T + 61T^{2} \) |
| 67 | \( 1 + 9.56T + 67T^{2} \) |
| 71 | \( 1 - 5.68T + 71T^{2} \) |
| 73 | \( 1 - 9.35T + 73T^{2} \) |
| 79 | \( 1 + 3.18T + 79T^{2} \) |
| 83 | \( 1 - 6.11T + 83T^{2} \) |
| 89 | \( 1 - 3.00T + 89T^{2} \) |
| 97 | \( 1 + 5.95T + 97T^{2} \) |
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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
−7.82728682139446109711073604101, −7.04588423249358087070234717471, −6.61023759735904531665979491875, −5.74995459313322142009610681621, −5.27035681808923100594049511313, −4.38606658281406077267189904251, −3.75809444165740236194297634313, −3.24083308163263132053387384031, −2.07443641160296704220364917616, −1.25326177512428589575105748125,
1.25326177512428589575105748125, 2.07443641160296704220364917616, 3.24083308163263132053387384031, 3.75809444165740236194297634313, 4.38606658281406077267189904251, 5.27035681808923100594049511313, 5.74995459313322142009610681621, 6.61023759735904531665979491875, 7.04588423249358087070234717471, 7.82728682139446109711073604101
