| L(s) = 1 | − 6.71·2-s + 9·3-s + 13.0·4-s − 60.4·6-s + 177.·7-s + 127.·8-s + 81·9-s − 121·11-s + 117.·12-s + 783.·13-s − 1.18e3·14-s − 1.27e3·16-s + 1.33e3·17-s − 543.·18-s − 915.·19-s + 1.59e3·21-s + 812.·22-s + 1.68e3·23-s + 1.14e3·24-s − 5.25e3·26-s + 729·27-s + 2.31e3·28-s + 1.72e3·29-s + 7.74e3·31-s + 4.46e3·32-s − 1.08e3·33-s − 8.96e3·34-s + ⋯ |
| L(s) = 1 | − 1.18·2-s + 0.577·3-s + 0.407·4-s − 0.685·6-s + 1.36·7-s + 0.702·8-s + 0.333·9-s − 0.301·11-s + 0.235·12-s + 1.28·13-s − 1.62·14-s − 1.24·16-s + 1.12·17-s − 0.395·18-s − 0.581·19-s + 0.789·21-s + 0.357·22-s + 0.664·23-s + 0.405·24-s − 1.52·26-s + 0.192·27-s + 0.557·28-s + 0.379·29-s + 1.44·31-s + 0.770·32-s − 0.174·33-s − 1.32·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.159175528\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.159175528\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 + 6.71T + 32T^{2} \) |
| 7 | \( 1 - 177.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 783.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.33e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 915.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.68e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.72e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.74e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.01e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.38e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.18e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.05e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.48e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.22e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.08e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.69e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.32e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.72e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.62e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 401.T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.34e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.68e5T + 8.58e9T^{2} \) |
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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
−9.278157056969762253215157604619, −8.517832589616944604359087232066, −8.051501437921471603828867826790, −7.44304012585180275156808129114, −6.16873943483054867403358449043, −4.91269070100956561361914999494, −4.08560659393155273810924482787, −2.68047964634945141536406107424, −1.46825893025117791109804199805, −0.899651680860863418024391609277,
0.899651680860863418024391609277, 1.46825893025117791109804199805, 2.68047964634945141536406107424, 4.08560659393155273810924482787, 4.91269070100956561361914999494, 6.16873943483054867403358449043, 7.44304012585180275156808129114, 8.051501437921471603828867826790, 8.517832589616944604359087232066, 9.278157056969762253215157604619
