| L(s) = 1 | − 2.18·2-s − 3-s + 2.76·4-s + 5-s + 2.18·6-s − 0.993·7-s − 1.67·8-s + 9-s − 2.18·10-s − 5.93·11-s − 2.76·12-s − 4.13·13-s + 2.16·14-s − 15-s − 1.87·16-s − 7.29·17-s − 2.18·18-s + 2.31·19-s + 2.76·20-s + 0.993·21-s + 12.9·22-s + 1.67·24-s + 25-s + 9.02·26-s − 27-s − 2.75·28-s − 5.22·29-s + ⋯ |
| L(s) = 1 | − 1.54·2-s − 0.577·3-s + 1.38·4-s + 0.447·5-s + 0.891·6-s − 0.375·7-s − 0.593·8-s + 0.333·9-s − 0.690·10-s − 1.78·11-s − 0.799·12-s − 1.14·13-s + 0.579·14-s − 0.258·15-s − 0.467·16-s − 1.76·17-s − 0.514·18-s + 0.530·19-s + 0.619·20-s + 0.216·21-s + 2.76·22-s + 0.342·24-s + 0.200·25-s + 1.76·26-s − 0.192·27-s − 0.519·28-s − 0.970·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.007530423797\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.007530423797\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 2.18T + 2T^{2} \) |
| 7 | \( 1 + 0.993T + 7T^{2} \) |
| 11 | \( 1 + 5.93T + 11T^{2} \) |
| 13 | \( 1 + 4.13T + 13T^{2} \) |
| 17 | \( 1 + 7.29T + 17T^{2} \) |
| 19 | \( 1 - 2.31T + 19T^{2} \) |
| 29 | \( 1 + 5.22T + 29T^{2} \) |
| 31 | \( 1 - 9.31T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 + 8.24T + 41T^{2} \) |
| 43 | \( 1 + 5.82T + 43T^{2} \) |
| 47 | \( 1 - 0.519T + 47T^{2} \) |
| 53 | \( 1 + 1.97T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 + 4.81T + 61T^{2} \) |
| 67 | \( 1 + 7.46T + 67T^{2} \) |
| 71 | \( 1 - 0.553T + 71T^{2} \) |
| 73 | \( 1 - 8.22T + 73T^{2} \) |
| 79 | \( 1 + 3.22T + 79T^{2} \) |
| 83 | \( 1 + 17.6T + 83T^{2} \) |
| 89 | \( 1 - 2.14T + 89T^{2} \) |
| 97 | \( 1 + 5.04T + 97T^{2} \) |
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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
−7.946085363269474942948439933924, −7.15905257498097455231783631453, −6.80576363861065997010270675742, −5.95971673101180633672968556445, −4.98839880083093175638982774198, −4.68727318614002459798499679263, −3.09236351342091887336604869422, −2.34642395075651839381835914926, −1.62625845788952760895733380410, −0.05383826302891702513378670181,
0.05383826302891702513378670181, 1.62625845788952760895733380410, 2.34642395075651839381835914926, 3.09236351342091887336604869422, 4.68727318614002459798499679263, 4.98839880083093175638982774198, 5.95971673101180633672968556445, 6.80576363861065997010270675742, 7.15905257498097455231783631453, 7.946085363269474942948439933924
