| L(s) = 1 | − 5.42·2-s + 21.4·4-s + 16.8·5-s − 7.69·7-s − 72.8·8-s − 91.3·10-s + 11·11-s + 24.8·13-s + 41.7·14-s + 223.·16-s + 15.9·17-s + 15.1·19-s + 360.·20-s − 59.6·22-s − 17.7·23-s + 158.·25-s − 134.·26-s − 164.·28-s + 128.·29-s + 219.·31-s − 630.·32-s − 86.4·34-s − 129.·35-s + 92.0·37-s − 82.1·38-s − 1.22e3·40-s + 459.·41-s + ⋯ |
| L(s) = 1 | − 1.91·2-s + 2.67·4-s + 1.50·5-s − 0.415·7-s − 3.21·8-s − 2.89·10-s + 0.301·11-s + 0.530·13-s + 0.797·14-s + 3.49·16-s + 0.227·17-s + 0.182·19-s + 4.03·20-s − 0.578·22-s − 0.160·23-s + 1.27·25-s − 1.01·26-s − 1.11·28-s + 0.823·29-s + 1.27·31-s − 3.48·32-s − 0.436·34-s − 0.626·35-s + 0.409·37-s − 0.350·38-s − 4.84·40-s + 1.75·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8639165479\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8639165479\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 + 5.42T + 8T^{2} \) |
| 5 | \( 1 - 16.8T + 125T^{2} \) |
| 7 | \( 1 + 7.69T + 343T^{2} \) |
| 13 | \( 1 - 24.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 15.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 15.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 17.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 128.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 219.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 92.0T + 5.06e4T^{2} \) |
| 41 | \( 1 - 459.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 64.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + 497.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 526.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 578.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 221.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 860.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 580.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 510.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.03e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 606.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 23.4T + 7.04e5T^{2} \) |
| 97 | \( 1 - 719.T + 9.12e5T^{2} \) |
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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
−13.36357204796690508666455791442, −11.98087991122259990076142989170, −10.74062031064241600045840746357, −9.886763094665814527666032787237, −9.278275354698618631066677104851, −8.177995380451782918339950309412, −6.70808927952813847529213459907, −5.93752445879423514060989834098, −2.64075537766779952728152551290, −1.20457236984055875159994227351,
1.20457236984055875159994227351, 2.64075537766779952728152551290, 5.93752445879423514060989834098, 6.70808927952813847529213459907, 8.177995380451782918339950309412, 9.278275354698618631066677104851, 9.886763094665814527666032787237, 10.74062031064241600045840746357, 11.98087991122259990076142989170, 13.36357204796690508666455791442
