| L(s) = 1 | + 0.149·2-s + 0.0395·3-s − 1.97·4-s + 5-s + 0.00590·6-s − 7-s − 0.593·8-s − 2.99·9-s + 0.149·10-s + 2.33·11-s − 0.0782·12-s − 5.45·13-s − 0.149·14-s + 0.0395·15-s + 3.86·16-s + 7.41·17-s − 0.447·18-s − 1.93·19-s − 1.97·20-s − 0.0395·21-s + 0.348·22-s + 3.95·23-s − 0.0234·24-s + 25-s − 0.814·26-s − 0.237·27-s + 1.97·28-s + ⋯ |
| L(s) = 1 | + 0.105·2-s + 0.0228·3-s − 0.988·4-s + 0.447·5-s + 0.00241·6-s − 0.377·7-s − 0.209·8-s − 0.999·9-s + 0.0471·10-s + 0.703·11-s − 0.0225·12-s − 1.51·13-s − 0.0398·14-s + 0.0102·15-s + 0.966·16-s + 1.79·17-s − 0.105·18-s − 0.444·19-s − 0.442·20-s − 0.00863·21-s + 0.0742·22-s + 0.824·23-s − 0.00479·24-s + 0.200·25-s − 0.159·26-s − 0.0456·27-s + 0.373·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 - 0.149T + 2T^{2} \) |
| 3 | \( 1 - 0.0395T + 3T^{2} \) |
| 11 | \( 1 - 2.33T + 11T^{2} \) |
| 13 | \( 1 + 5.45T + 13T^{2} \) |
| 17 | \( 1 - 7.41T + 17T^{2} \) |
| 19 | \( 1 + 1.93T + 19T^{2} \) |
| 23 | \( 1 - 3.95T + 23T^{2} \) |
| 29 | \( 1 - 8.72T + 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 + 4.27T + 37T^{2} \) |
| 41 | \( 1 + 8.49T + 41T^{2} \) |
| 43 | \( 1 + 3.87T + 43T^{2} \) |
| 47 | \( 1 + 6.50T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 - 9.88T + 59T^{2} \) |
| 61 | \( 1 - 2.58T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 - 3.39T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 - 17.6T + 79T^{2} \) |
| 83 | \( 1 + 14.8T + 83T^{2} \) |
| 89 | \( 1 + 4.12T + 89T^{2} \) |
| 97 | \( 1 + 6.76T + 97T^{2} \) |
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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
−7.43825015078326165583332665332, −6.81901626079978444390671923280, −5.94279570116182783534477028981, −5.16272538142468433823798685349, −5.03021635600264255266204556918, −3.73678105380237114731244623021, −3.26341120180234327918122071297, −2.35264108274328719766802507628, −1.10080443522256767545345551178, 0,
1.10080443522256767545345551178, 2.35264108274328719766802507628, 3.26341120180234327918122071297, 3.73678105380237114731244623021, 5.03021635600264255266204556918, 5.16272538142468433823798685349, 5.94279570116182783534477028981, 6.81901626079978444390671923280, 7.43825015078326165583332665332
