| L(s) = 1 | + 4.17·2-s + 0.180·3-s + 9.46·4-s + 12.4·5-s + 0.753·6-s + 6.13·8-s − 26.9·9-s + 52.1·10-s − 12.2·11-s + 1.70·12-s − 53.4·13-s + 2.24·15-s − 50.1·16-s − 2.95·17-s − 112.·18-s − 161.·19-s + 118.·20-s − 51.0·22-s + 23·23-s + 1.10·24-s + 30.6·25-s − 223.·26-s − 9.73·27-s + 96.2·29-s + 9.40·30-s − 87.0·31-s − 258.·32-s + ⋯ |
| L(s) = 1 | + 1.47·2-s + 0.0347·3-s + 1.18·4-s + 1.11·5-s + 0.0512·6-s + 0.270·8-s − 0.998·9-s + 1.64·10-s − 0.334·11-s + 0.0410·12-s − 1.13·13-s + 0.0387·15-s − 0.783·16-s − 0.0421·17-s − 1.47·18-s − 1.95·19-s + 1.32·20-s − 0.494·22-s + 0.208·23-s + 0.00940·24-s + 0.245·25-s − 1.68·26-s − 0.0693·27-s + 0.616·29-s + 0.0572·30-s − 0.504·31-s − 1.42·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 - 4.17T + 8T^{2} \) |
| 3 | \( 1 - 0.180T + 27T^{2} \) |
| 5 | \( 1 - 12.4T + 125T^{2} \) |
| 11 | \( 1 + 12.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 53.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 2.95T + 4.91e3T^{2} \) |
| 19 | \( 1 + 161.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 96.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 87.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 75.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 153.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 99.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + 322.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 565.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 537.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 433.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 588.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.05e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 639.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 170.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.03e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 476.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 146.T + 9.12e5T^{2} \) |
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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
−9.052941209284651304996809785262, −8.245142196458444330456303969910, −6.87959931658391822610861598732, −6.25758599487110008843509033943, −5.43611411951196814128492405597, −4.90935888331845727787436589061, −3.80254537595903158303206875937, −2.58692259876906301451275593089, −2.16209556582394491228220225554, 0,
2.16209556582394491228220225554, 2.58692259876906301451275593089, 3.80254537595903158303206875937, 4.90935888331845727787436589061, 5.43611411951196814128492405597, 6.25758599487110008843509033943, 6.87959931658391822610861598732, 8.245142196458444330456303969910, 9.052941209284651304996809785262
