| L(s) = 1 | − 85.0·2-s + 243·3-s + 5.17e3·4-s − 2.06e4·6-s + 3.87e4·7-s − 2.66e5·8-s + 5.90e4·9-s − 9.78e5·11-s + 1.25e6·12-s + 1.02e6·13-s − 3.29e6·14-s + 1.20e7·16-s + 4.83e6·17-s − 5.01e6·18-s + 6.37e6·19-s + 9.41e6·21-s + 8.31e7·22-s + 1.55e7·23-s − 6.46e7·24-s − 8.74e7·26-s + 1.43e7·27-s + 2.00e8·28-s + 3.13e7·29-s − 7.65e7·31-s − 4.76e8·32-s − 2.37e8·33-s − 4.10e8·34-s + ⋯ |
| L(s) = 1 | − 1.87·2-s + 0.577·3-s + 2.52·4-s − 1.08·6-s + 0.871·7-s − 2.87·8-s + 0.333·9-s − 1.83·11-s + 1.46·12-s + 0.768·13-s − 1.63·14-s + 2.86·16-s + 0.825·17-s − 0.626·18-s + 0.590·19-s + 0.502·21-s + 3.44·22-s + 0.504·23-s − 1.65·24-s − 1.44·26-s + 0.192·27-s + 2.20·28-s + 0.283·29-s − 0.480·31-s − 2.51·32-s − 1.05·33-s − 1.55·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.138602175\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.138602175\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 243T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 85.0T + 2.04e3T^{2} \) |
| 7 | \( 1 - 3.87e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 9.78e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 1.02e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 4.83e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 6.37e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 1.55e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 3.13e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + 7.65e7T + 2.54e16T^{2} \) |
| 37 | \( 1 - 2.61e7T + 1.77e17T^{2} \) |
| 41 | \( 1 + 6.83e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 6.31e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 5.74e8T + 2.47e18T^{2} \) |
| 53 | \( 1 - 5.41e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 3.70e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 8.47e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 9.26e8T + 1.22e20T^{2} \) |
| 71 | \( 1 + 2.77e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 7.87e9T + 3.13e20T^{2} \) |
| 79 | \( 1 - 1.41e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 5.50e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 8.53e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 9.23e10T + 7.15e21T^{2} \) |
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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
−11.75185511586512844987112335055, −10.70181736674416049218261559901, −9.986501059563662113074145738363, −8.661528897080771465908511771666, −8.024240995084528519220913131843, −7.19886588226142543555120045528, −5.43396604314543649345833762739, −3.05013850711982061082382991580, −1.89786838563466605248177907396, −0.75710856000532927437249480332,
0.75710856000532927437249480332, 1.89786838563466605248177907396, 3.05013850711982061082382991580, 5.43396604314543649345833762739, 7.19886588226142543555120045528, 8.024240995084528519220913131843, 8.661528897080771465908511771666, 9.986501059563662113074145738363, 10.70181736674416049218261559901, 11.75185511586512844987112335055
