| L(s) = 1 | − 5.32·2-s + 8.14·3-s + 20.3·4-s − 13.9·5-s − 43.3·6-s − 65.7·8-s + 39.2·9-s + 74.0·10-s − 20.6·11-s + 165.·12-s + 52.3·13-s − 113.·15-s + 187.·16-s + 18.4·17-s − 209.·18-s − 147.·19-s − 282.·20-s + 110.·22-s + 6.30·23-s − 535.·24-s + 68.3·25-s − 278.·26-s + 99.9·27-s − 136.·29-s + 602.·30-s + 43.8·31-s − 471.·32-s + ⋯ |
| L(s) = 1 | − 1.88·2-s + 1.56·3-s + 2.54·4-s − 1.24·5-s − 2.94·6-s − 2.90·8-s + 1.45·9-s + 2.34·10-s − 0.566·11-s + 3.98·12-s + 1.11·13-s − 1.94·15-s + 2.92·16-s + 0.262·17-s − 2.73·18-s − 1.78·19-s − 3.16·20-s + 1.06·22-s + 0.0571·23-s − 4.55·24-s + 0.546·25-s − 2.10·26-s + 0.712·27-s − 0.875·29-s + 3.66·30-s + 0.254·31-s − 2.60·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 2 | \( 1 + 5.32T + 8T^{2} \) |
| 3 | \( 1 - 8.14T + 27T^{2} \) |
| 5 | \( 1 + 13.9T + 125T^{2} \) |
| 11 | \( 1 + 20.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 52.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 18.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 147.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 6.30T + 1.21e4T^{2} \) |
| 29 | \( 1 + 136.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 43.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 173.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 121.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 474.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 167.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 531.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 646.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 125.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 266.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 409.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 338.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 93.6T + 4.93e5T^{2} \) |
| 83 | \( 1 - 785.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.36e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.40e3T + 9.12e5T^{2} \) |
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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
−10.50791075033185192856087943592, −9.273451229634415636873675384755, −8.683534087028564780113059704417, −7.987823366577196654427631650803, −7.58785807744922345404788536324, −6.39325970422094511054732694715, −3.94657753471245380230401734190, −2.90700685481000758450343605504, −1.68987134464273166139876651997, 0,
1.68987134464273166139876651997, 2.90700685481000758450343605504, 3.94657753471245380230401734190, 6.39325970422094511054732694715, 7.58785807744922345404788536324, 7.987823366577196654427631650803, 8.683534087028564780113059704417, 9.273451229634415636873675384755, 10.50791075033185192856087943592
