| L(s) = 1 | + 2.54·2-s − 2.04·3-s + 4.46·4-s − 5-s − 5.19·6-s + 2.42·7-s + 6.25·8-s + 1.18·9-s − 2.54·10-s + 2.68·11-s − 9.12·12-s + 4.12·13-s + 6.16·14-s + 2.04·15-s + 6.97·16-s + 0.0217·17-s + 3.01·18-s − 5.75·19-s − 4.46·20-s − 4.96·21-s + 6.81·22-s + 3.09·23-s − 12.7·24-s + 25-s + 10.4·26-s + 3.71·27-s + 10.8·28-s + ⋯ |
| L(s) = 1 | + 1.79·2-s − 1.18·3-s + 2.23·4-s − 0.447·5-s − 2.12·6-s + 0.916·7-s + 2.21·8-s + 0.394·9-s − 0.803·10-s + 0.808·11-s − 2.63·12-s + 1.14·13-s + 1.64·14-s + 0.528·15-s + 1.74·16-s + 0.00526·17-s + 0.709·18-s − 1.31·19-s − 0.997·20-s − 1.08·21-s + 1.45·22-s + 0.644·23-s − 2.61·24-s + 0.200·25-s + 2.05·26-s + 0.714·27-s + 2.04·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.959295285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.959295285\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 - 2.54T + 2T^{2} \) |
| 3 | \( 1 + 2.04T + 3T^{2} \) |
| 7 | \( 1 - 2.42T + 7T^{2} \) |
| 11 | \( 1 - 2.68T + 11T^{2} \) |
| 13 | \( 1 - 4.12T + 13T^{2} \) |
| 17 | \( 1 - 0.0217T + 17T^{2} \) |
| 19 | \( 1 + 5.75T + 19T^{2} \) |
| 23 | \( 1 - 3.09T + 23T^{2} \) |
| 29 | \( 1 + 1.01T + 29T^{2} \) |
| 31 | \( 1 + 3.56T + 31T^{2} \) |
| 37 | \( 1 - 0.119T + 37T^{2} \) |
| 41 | \( 1 - 5.35T + 41T^{2} \) |
| 43 | \( 1 - 8.29T + 43T^{2} \) |
| 47 | \( 1 - 9.24T + 47T^{2} \) |
| 53 | \( 1 - 7.85T + 53T^{2} \) |
| 59 | \( 1 + 4.32T + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 - 8.04T + 67T^{2} \) |
| 71 | \( 1 - 8.39T + 71T^{2} \) |
| 73 | \( 1 + 5.50T + 73T^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 + 17.0T + 83T^{2} \) |
| 89 | \( 1 - 4.17T + 89T^{2} \) |
| 97 | \( 1 + 12.5T + 97T^{2} \) |
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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
−9.046724387451560719778671189493, −8.165258762191436319966552075778, −7.06325279325326764670965166300, −6.48349469904942883656072868357, −5.74486969060195994749091088205, −5.19082599597629530445000423531, −4.19655931495214201861895447071, −3.89738000805673478526192832637, −2.46905600268031013139446723099, −1.18128287814809153616398712333,
1.18128287814809153616398712333, 2.46905600268031013139446723099, 3.89738000805673478526192832637, 4.19655931495214201861895447071, 5.19082599597629530445000423531, 5.74486969060195994749091088205, 6.48349469904942883656072868357, 7.06325279325326764670965166300, 8.165258762191436319966552075778, 9.046724387451560719778671189493
