| L(s) = 1 | − 6.55·2-s + 9·3-s + 11.0·4-s − 59.0·6-s − 146.·7-s + 137.·8-s + 81·9-s − 121·11-s + 99.1·12-s − 170.·13-s + 960.·14-s − 1.25e3·16-s − 1.56e3·17-s − 531.·18-s + 569.·19-s − 1.31e3·21-s + 793.·22-s − 3.15e3·23-s + 1.23e3·24-s + 1.11e3·26-s + 729·27-s − 1.61e3·28-s + 3.98e3·29-s + 2.99e3·31-s + 3.82e3·32-s − 1.08e3·33-s + 1.02e4·34-s + ⋯ |
| L(s) = 1 | − 1.15·2-s + 0.577·3-s + 0.344·4-s − 0.669·6-s − 1.12·7-s + 0.760·8-s + 0.333·9-s − 0.301·11-s + 0.198·12-s − 0.279·13-s + 1.31·14-s − 1.22·16-s − 1.31·17-s − 0.386·18-s + 0.362·19-s − 0.652·21-s + 0.349·22-s − 1.24·23-s + 0.438·24-s + 0.323·26-s + 0.192·27-s − 0.389·28-s + 0.879·29-s + 0.558·31-s + 0.661·32-s − 0.174·33-s + 1.52·34-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
| \(L(3)\) |
\(\approx\) |
\(0.5010688061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5010688061\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 6.55T + 32T^{2} \) |
| 7 | \( 1 + 146.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 170.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.56e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 569.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.15e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.98e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.99e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.85e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.20e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.38e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 6.85e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.83e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 9.64e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.11e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.34e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.26e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.43e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.89e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.64e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.03e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 7.58e4T + 8.58e9T^{2} \) |
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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
−9.542745238201600010050308100414, −8.626471160625154004603973785007, −8.102718313577529306531088851740, −7.04229408658193004512435942869, −6.43393259516918565755940309837, −4.95152447542382480304315965095, −3.90476392203606472018964619067, −2.75814671586818336309329274646, −1.75103608849350461757826734523, −0.36346721324190111573002025139,
0.36346721324190111573002025139, 1.75103608849350461757826734523, 2.75814671586818336309329274646, 3.90476392203606472018964619067, 4.95152447542382480304315965095, 6.43393259516918565755940309837, 7.04229408658193004512435942869, 8.102718313577529306531088851740, 8.626471160625154004603973785007, 9.542745238201600010050308100414
