| L(s) = 1 | − 2-s + 0.622·3-s + 4-s + 5-s − 0.622·6-s + 3.30·7-s − 8-s − 2.61·9-s − 10-s + 1.88·11-s + 0.622·12-s + 1.73·13-s − 3.30·14-s + 0.622·15-s + 16-s + 5.95·17-s + 2.61·18-s + 2.09·19-s + 20-s + 2.05·21-s − 1.88·22-s + 5.93·23-s − 0.622·24-s + 25-s − 1.73·26-s − 3.49·27-s + 3.30·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.359·3-s + 0.5·4-s + 0.447·5-s − 0.254·6-s + 1.24·7-s − 0.353·8-s − 0.870·9-s − 0.316·10-s + 0.567·11-s + 0.179·12-s + 0.481·13-s − 0.883·14-s + 0.160·15-s + 0.250·16-s + 1.44·17-s + 0.615·18-s + 0.479·19-s + 0.223·20-s + 0.449·21-s − 0.401·22-s + 1.23·23-s − 0.127·24-s + 0.200·25-s − 0.340·26-s − 0.672·27-s + 0.624·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.221352618\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.221352618\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 - 0.622T + 3T^{2} \) |
| 7 | \( 1 - 3.30T + 7T^{2} \) |
| 11 | \( 1 - 1.88T + 11T^{2} \) |
| 13 | \( 1 - 1.73T + 13T^{2} \) |
| 17 | \( 1 - 5.95T + 17T^{2} \) |
| 19 | \( 1 - 2.09T + 19T^{2} \) |
| 23 | \( 1 - 5.93T + 23T^{2} \) |
| 29 | \( 1 - 4.40T + 29T^{2} \) |
| 31 | \( 1 - 2.17T + 31T^{2} \) |
| 37 | \( 1 + 2.94T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 + 6.51T + 43T^{2} \) |
| 47 | \( 1 + 2.61T + 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 - 3.12T + 59T^{2} \) |
| 61 | \( 1 + 8.50T + 61T^{2} \) |
| 67 | \( 1 - 5.94T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 - 3.03T + 73T^{2} \) |
| 79 | \( 1 - 6.13T + 79T^{2} \) |
| 83 | \( 1 + 9.86T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
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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
−8.609436262162233199596375129435, −7.919431823071347957813712919248, −7.16910531779895136046864552580, −6.32396194169703278628176922474, −5.44474730791041288996166527573, −4.91772315399770362420881710812, −3.55655255006759599234652498853, −2.87375631157633576514451683825, −1.73029958987328197491649812994, −1.02885049409031313938020464006,
1.02885049409031313938020464006, 1.73029958987328197491649812994, 2.87375631157633576514451683825, 3.55655255006759599234652498853, 4.91772315399770362420881710812, 5.44474730791041288996166527573, 6.32396194169703278628176922474, 7.16910531779895136046864552580, 7.919431823071347957813712919248, 8.609436262162233199596375129435
