| L(s) = 1 | − 0.347·2-s + 0.684·3-s − 0.879·4-s − 1.96·5-s − 0.237·6-s − 7-s + 0.652·8-s − 0.532·9-s + 0.684·10-s − 11-s − 0.601·12-s − 1.73·13-s + 0.347·14-s − 1.34·15-s + 0.652·16-s + 1.28·17-s + 0.184·18-s + 1.73·20-s − 0.684·21-s + 0.347·22-s + 23-s + 0.446·24-s + 2.87·25-s + 0.601·26-s − 1.04·27-s + 0.879·28-s − 1.87·29-s + ⋯ |
| L(s) = 1 | − 0.347·2-s + 0.684·3-s − 0.879·4-s − 1.96·5-s − 0.237·6-s − 7-s + 0.652·8-s − 0.532·9-s + 0.684·10-s − 11-s − 0.601·12-s − 1.73·13-s + 0.347·14-s − 1.34·15-s + 0.652·16-s + 1.28·17-s + 0.184·18-s + 1.73·20-s − 0.684·21-s + 0.347·22-s + 23-s + 0.446·24-s + 2.87·25-s + 0.601·26-s − 1.04·27-s + 0.879·28-s − 1.87·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2889882356\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2889882356\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 0.347T + T^{2} \) |
| 3 | \( 1 - 0.684T + T^{2} \) |
| 5 | \( 1 + 1.96T + T^{2} \) |
| 13 | \( 1 + 1.73T + T^{2} \) |
| 17 | \( 1 - 1.28T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( 1 + 1.87T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 1.28T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 1.53T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 0.347T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - 0.684T + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.755917474110207569790927109987, −8.028799916106622702207214852000, −7.47897272234068054968450581961, −7.19620212817481146050890344799, −5.50103067224158058582924411314, −4.95280154293824044924145595704, −3.89051929061376608274061306591, −3.36114866313686682902960155952, −2.63768066237565615846388638570, −0.45422024122663926346917650619,
0.45422024122663926346917650619, 2.63768066237565615846388638570, 3.36114866313686682902960155952, 3.89051929061376608274061306591, 4.95280154293824044924145595704, 5.50103067224158058582924411314, 7.19620212817481146050890344799, 7.47897272234068054968450581961, 8.028799916106622702207214852000, 8.755917474110207569790927109987
