| L(s) = 1 | − 17.3·3-s − 25·5-s − 49·7-s + 56.5·9-s − 151.·11-s − 1.05e3·13-s + 432.·15-s + 1.19e3·17-s − 2.18e3·19-s + 848.·21-s + 4.20e3·23-s + 625·25-s + 3.22e3·27-s − 657.·29-s + 1.07e3·31-s + 2.63e3·33-s + 1.22e3·35-s + 959.·37-s + 1.82e4·39-s + 9.47e3·41-s + 1.74e4·43-s − 1.41e3·45-s + 1.11e4·47-s + 2.40e3·49-s − 2.06e4·51-s − 3.21e4·53-s + 3.79e3·55-s + ⋯ |
| L(s) = 1 | − 1.11·3-s − 0.447·5-s − 0.377·7-s + 0.232·9-s − 0.378·11-s − 1.72·13-s + 0.496·15-s + 0.999·17-s − 1.39·19-s + 0.419·21-s + 1.65·23-s + 0.200·25-s + 0.851·27-s − 0.145·29-s + 0.200·31-s + 0.420·33-s + 0.169·35-s + 0.115·37-s + 1.91·39-s + 0.880·41-s + 1.43·43-s − 0.104·45-s + 0.737·47-s + 0.142·49-s − 1.11·51-s − 1.57·53-s + 0.169·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.6586158466\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6586158466\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 25T \) |
| 7 | \( 1 + 49T \) |
| good | 3 | \( 1 + 17.3T + 243T^{2} \) |
| 11 | \( 1 + 151.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.05e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.19e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.18e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.20e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 657.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.07e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 959.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 9.47e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.74e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.11e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.21e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.65e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.88e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.79e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.73e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.84e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.00e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.03e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.23e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.92e4T + 8.58e9T^{2} \) |
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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
−12.36814505499458194083050551361, −11.20074786653771100726750259237, −10.43400527537175100412866217267, −9.265262993677098725960564570513, −7.78162931112408169094601657639, −6.76002255588236951831229219598, −5.53709566875505356980965764651, −4.55708218285771078839358742975, −2.76433384050796553801306472650, −0.54918156597481852004264033632,
0.54918156597481852004264033632, 2.76433384050796553801306472650, 4.55708218285771078839358742975, 5.53709566875505356980965764651, 6.76002255588236951831229219598, 7.78162931112408169094601657639, 9.265262993677098725960564570513, 10.43400527537175100412866217267, 11.20074786653771100726750259237, 12.36814505499458194083050551361
