L(s) = 1 | − 2-s + 0.576·3-s + 4-s + 5-s − 0.576·6-s − 4.86·7-s − 8-s − 2.66·9-s − 10-s − 5.36·11-s + 0.576·12-s − 3.85·13-s + 4.86·14-s + 0.576·15-s + 16-s + 1.40·17-s + 2.66·18-s + 20-s − 2.80·21-s + 5.36·22-s + 5.33·23-s − 0.576·24-s + 25-s + 3.85·26-s − 3.26·27-s − 4.86·28-s − 3.63·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.332·3-s + 0.5·4-s + 0.447·5-s − 0.235·6-s − 1.83·7-s − 0.353·8-s − 0.889·9-s − 0.316·10-s − 1.61·11-s + 0.166·12-s − 1.06·13-s + 1.30·14-s + 0.148·15-s + 0.250·16-s + 0.340·17-s + 0.628·18-s + 0.223·20-s − 0.611·21-s + 1.14·22-s + 1.11·23-s − 0.117·24-s + 0.200·25-s + 0.756·26-s − 0.628·27-s − 0.919·28-s − 0.674·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5285232511\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5285232511\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 - 0.576T + 3T^{2} \) |
| 7 | \( 1 + 4.86T + 7T^{2} \) |
| 11 | \( 1 + 5.36T + 11T^{2} \) |
| 13 | \( 1 + 3.85T + 13T^{2} \) |
| 17 | \( 1 - 1.40T + 17T^{2} \) |
| 23 | \( 1 - 5.33T + 23T^{2} \) |
| 29 | \( 1 + 3.63T + 29T^{2} \) |
| 31 | \( 1 - 8.20T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 + 1.90T + 41T^{2} \) |
| 43 | \( 1 - 1.47T + 43T^{2} \) |
| 47 | \( 1 + 3.12T + 47T^{2} \) |
| 53 | \( 1 + 1.80T + 53T^{2} \) |
| 59 | \( 1 + 7.32T + 59T^{2} \) |
| 61 | \( 1 - 8.11T + 61T^{2} \) |
| 67 | \( 1 - 8.44T + 67T^{2} \) |
| 71 | \( 1 - 6.02T + 71T^{2} \) |
| 73 | \( 1 + 1.27T + 73T^{2} \) |
| 79 | \( 1 + 16.6T + 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 - 9.60T + 89T^{2} \) |
| 97 | \( 1 + 6.07T + 97T^{2} \) |
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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
−8.661696632667592155690919426023, −7.87113003335339866654162733287, −7.12523295900580360836218109052, −6.47802932390341702210621116129, −5.61738199625806106697744276334, −5.02770001613026743054792738164, −3.38062219645190116876449672461, −2.88144184642024062126933167055, −2.26275977496670493294508802990, −0.42506709264682214448435957834,
0.42506709264682214448435957834, 2.26275977496670493294508802990, 2.88144184642024062126933167055, 3.38062219645190116876449672461, 5.02770001613026743054792738164, 5.61738199625806106697744276334, 6.47802932390341702210621116129, 7.12523295900580360836218109052, 7.87113003335339866654162733287, 8.661696632667592155690919426023