| L(s) = 1 | + 0.315·2-s − 7.90·4-s + 5.90·5-s − 7·7-s − 5.02·8-s + 1.86·10-s − 11·11-s + 2.91·13-s − 2.21·14-s + 61.6·16-s − 60.0·17-s + 69.8·19-s − 46.6·20-s − 3.47·22-s − 120.·23-s − 90.0·25-s + 0.919·26-s + 55.3·28-s + 174.·29-s + 44.5·31-s + 59.6·32-s − 18.9·34-s − 41.3·35-s − 271.·37-s + 22.0·38-s − 29.6·40-s + 355.·41-s + ⋯ |
| L(s) = 1 | + 0.111·2-s − 0.987·4-s + 0.528·5-s − 0.377·7-s − 0.221·8-s + 0.0590·10-s − 0.301·11-s + 0.0621·13-s − 0.0422·14-s + 0.962·16-s − 0.856·17-s + 0.843·19-s − 0.521·20-s − 0.0336·22-s − 1.09·23-s − 0.720·25-s + 0.00693·26-s + 0.373·28-s + 1.11·29-s + 0.258·31-s + 0.329·32-s − 0.0956·34-s − 0.199·35-s − 1.20·37-s + 0.0942·38-s − 0.117·40-s + 1.35·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.436587059\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.436587059\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 0.315T + 8T^{2} \) |
| 5 | \( 1 - 5.90T + 125T^{2} \) |
| 13 | \( 1 - 2.91T + 2.19e3T^{2} \) |
| 17 | \( 1 + 60.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 69.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 120.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 174.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 44.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 271.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 355.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 545.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 413.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 709.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 358.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 574.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 457.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 87.8T + 3.57e5T^{2} \) |
| 73 | \( 1 - 403.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 195.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.40e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.25e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.77e3T + 9.12e5T^{2} \) |
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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
−9.922674286114297483593078207728, −9.301901727437274919716694085496, −8.467171283841765216067512600360, −7.54424990042543949182251310580, −6.30722071426424992420266364336, −5.55039111884068068099827661104, −4.55962583521609171909128193281, −3.59036851815733776248908678962, −2.29194172114540316248114590212, −0.67767443125240349721625218839,
0.67767443125240349721625218839, 2.29194172114540316248114590212, 3.59036851815733776248908678962, 4.55962583521609171909128193281, 5.55039111884068068099827661104, 6.30722071426424992420266364336, 7.54424990042543949182251310580, 8.467171283841765216067512600360, 9.301901727437274919716694085496, 9.922674286114297483593078207728
