| L(s) = 1 | + 4·2-s + 21.8·3-s + 16·4-s + 25·5-s + 87.3·6-s + 7.44·7-s + 64·8-s + 234.·9-s + 100·10-s − 441.·11-s + 349.·12-s + 785.·13-s + 29.7·14-s + 546.·15-s + 256·16-s + 632.·17-s + 936.·18-s + 2.08e3·19-s + 400·20-s + 162.·21-s − 1.76e3·22-s + 529·23-s + 1.39e3·24-s + 625·25-s + 3.14e3·26-s − 191.·27-s + 119.·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.40·3-s + 0.5·4-s + 0.447·5-s + 0.990·6-s + 0.0574·7-s + 0.353·8-s + 0.963·9-s + 0.316·10-s − 1.10·11-s + 0.700·12-s + 1.28·13-s + 0.0406·14-s + 0.626·15-s + 0.250·16-s + 0.530·17-s + 0.681·18-s + 1.32·19-s + 0.223·20-s + 0.0804·21-s − 0.778·22-s + 0.208·23-s + 0.495·24-s + 0.200·25-s + 0.911·26-s − 0.0505·27-s + 0.0287·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.818431061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.818431061\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 5 | \( 1 - 25T \) |
| 23 | \( 1 - 529T \) |
| good | 3 | \( 1 - 21.8T + 243T^{2} \) |
| 7 | \( 1 - 7.44T + 1.68e4T^{2} \) |
| 11 | \( 1 + 441.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 785.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 632.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.08e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 3.12e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 416.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 744.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.01e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.05e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.74e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.03e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.06e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 6.55e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.95e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.42e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.13e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.93e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.25e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.29e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.49e4T + 8.58e9T^{2} \) |
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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
−11.37437341059552290689845106376, −10.30855824694099653843263941362, −9.331555022048874901314298940904, −8.260312688260493189932073167819, −7.52676818406350725735966790125, −6.11323985663594085926122001787, −4.98126041627884667342363879708, −3.48854632119274997277390113189, −2.78985778473477628799191337104, −1.45121459314454758111087132877,
1.45121459314454758111087132877, 2.78985778473477628799191337104, 3.48854632119274997277390113189, 4.98126041627884667342363879708, 6.11323985663594085926122001787, 7.52676818406350725735966790125, 8.260312688260493189932073167819, 9.331555022048874901314298940904, 10.30855824694099653843263941362, 11.37437341059552290689845106376
