| L(s) = 1 | + 2.04·2-s + 3-s + 2.16·4-s + 4.10·5-s + 2.04·6-s − 7-s + 0.333·8-s + 9-s + 8.38·10-s + 1.94·11-s + 2.16·12-s − 3.26·13-s − 2.04·14-s + 4.10·15-s − 3.64·16-s + 2.41·17-s + 2.04·18-s + 2.82·19-s + 8.88·20-s − 21-s + 3.97·22-s + 0.499·23-s + 0.333·24-s + 11.8·25-s − 6.66·26-s + 27-s − 2.16·28-s + ⋯ |
| L(s) = 1 | + 1.44·2-s + 0.577·3-s + 1.08·4-s + 1.83·5-s + 0.833·6-s − 0.377·7-s + 0.118·8-s + 0.333·9-s + 2.65·10-s + 0.587·11-s + 0.624·12-s − 0.906·13-s − 0.545·14-s + 1.06·15-s − 0.911·16-s + 0.585·17-s + 0.480·18-s + 0.648·19-s + 1.98·20-s − 0.218·21-s + 0.847·22-s + 0.104·23-s + 0.0681·24-s + 2.37·25-s − 1.30·26-s + 0.192·27-s − 0.408·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.195046004\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.195046004\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 - 2.04T + 2T^{2} \) |
| 5 | \( 1 - 4.10T + 5T^{2} \) |
| 11 | \( 1 - 1.94T + 11T^{2} \) |
| 13 | \( 1 + 3.26T + 13T^{2} \) |
| 17 | \( 1 - 2.41T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 0.499T + 23T^{2} \) |
| 29 | \( 1 - 3.85T + 29T^{2} \) |
| 31 | \( 1 + 5.31T + 31T^{2} \) |
| 37 | \( 1 + 6.29T + 37T^{2} \) |
| 41 | \( 1 - 7.77T + 41T^{2} \) |
| 43 | \( 1 - 7.67T + 43T^{2} \) |
| 47 | \( 1 - 13.4T + 47T^{2} \) |
| 53 | \( 1 - 1.71T + 53T^{2} \) |
| 59 | \( 1 + 6.54T + 59T^{2} \) |
| 61 | \( 1 - 7.95T + 61T^{2} \) |
| 67 | \( 1 + 0.961T + 67T^{2} \) |
| 71 | \( 1 - 1.89T + 71T^{2} \) |
| 73 | \( 1 + 0.962T + 73T^{2} \) |
| 79 | \( 1 - 0.315T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 + 8.33T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−7.42422182539810875823905266456, −6.97191564008807411150858098461, −6.15663284185745628173035469708, −5.65978984876600423031834956201, −5.14706800564978186108556353801, −4.31753929653653787669837365211, −3.49227861139611139058051160880, −2.67457621483909316208409121200, −2.27305141541536132276267349760, −1.19087158715568859282557490113,
1.19087158715568859282557490113, 2.27305141541536132276267349760, 2.67457621483909316208409121200, 3.49227861139611139058051160880, 4.31753929653653787669837365211, 5.14706800564978186108556353801, 5.65978984876600423031834956201, 6.15663284185745628173035469708, 6.97191564008807411150858098461, 7.42422182539810875823905266456
