| L(s) = 1 | − 2-s − 1.71·3-s + 4-s + 1.96·5-s + 1.71·6-s + 0.664·7-s − 8-s − 0.0513·9-s − 1.96·10-s + 5.44·11-s − 1.71·12-s − 0.0515·13-s − 0.664·14-s − 3.37·15-s + 16-s − 7.45·17-s + 0.0513·18-s − 19-s + 1.96·20-s − 1.14·21-s − 5.44·22-s − 0.569·23-s + 1.71·24-s − 1.13·25-s + 0.0515·26-s + 5.23·27-s + 0.664·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.991·3-s + 0.5·4-s + 0.879·5-s + 0.701·6-s + 0.251·7-s − 0.353·8-s − 0.0171·9-s − 0.621·10-s + 1.64·11-s − 0.495·12-s − 0.0143·13-s − 0.177·14-s − 0.871·15-s + 0.250·16-s − 1.80·17-s + 0.0121·18-s − 0.229·19-s + 0.439·20-s − 0.249·21-s − 1.16·22-s − 0.118·23-s + 0.350·24-s − 0.226·25-s + 0.0101·26-s + 1.00·27-s + 0.125·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 + 1.71T + 3T^{2} \) |
| 5 | \( 1 - 1.96T + 5T^{2} \) |
| 7 | \( 1 - 0.664T + 7T^{2} \) |
| 11 | \( 1 - 5.44T + 11T^{2} \) |
| 13 | \( 1 + 0.0515T + 13T^{2} \) |
| 17 | \( 1 + 7.45T + 17T^{2} \) |
| 23 | \( 1 + 0.569T + 23T^{2} \) |
| 29 | \( 1 + 2.76T + 29T^{2} \) |
| 31 | \( 1 - 4.18T + 31T^{2} \) |
| 37 | \( 1 - 3.80T + 37T^{2} \) |
| 41 | \( 1 - 3.76T + 41T^{2} \) |
| 43 | \( 1 - 5.34T + 43T^{2} \) |
| 47 | \( 1 - 2.90T + 47T^{2} \) |
| 53 | \( 1 + 4.82T + 53T^{2} \) |
| 59 | \( 1 + 0.401T + 59T^{2} \) |
| 61 | \( 1 + 7.28T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 + 15.3T + 71T^{2} \) |
| 73 | \( 1 + 0.623T + 73T^{2} \) |
| 79 | \( 1 + 3.96T + 79T^{2} \) |
| 83 | \( 1 - 7.33T + 83T^{2} \) |
| 89 | \( 1 - 11.5T + 89T^{2} \) |
| 97 | \( 1 + 1.39T + 97T^{2} \) |
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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
−7.36513223907952866052888468715, −6.55022137264758150917006636797, −6.23934672790005764495654920661, −5.74444774474351837808905416934, −4.66979482961666937725899958373, −4.13828800656914265376131885833, −2.82883798211946360620274932507, −1.93395474034878935681888361170, −1.19260927561614373807516026961, 0,
1.19260927561614373807516026961, 1.93395474034878935681888361170, 2.82883798211946360620274932507, 4.13828800656914265376131885833, 4.66979482961666937725899958373, 5.74444774474351837808905416934, 6.23934672790005764495654920661, 6.55022137264758150917006636797, 7.36513223907952866052888468715
