| L(s) = 1 | + 29.8·2-s − 243·3-s − 1.15e3·4-s − 7.26e3·6-s + 6.01e4·7-s − 9.57e4·8-s + 5.90e4·9-s − 7.70e5·11-s + 2.80e5·12-s + 2.93e5·13-s + 1.79e6·14-s − 4.93e5·16-s + 4.97e5·17-s + 1.76e6·18-s − 1.40e7·19-s − 1.46e7·21-s − 2.30e7·22-s − 5.26e6·23-s + 2.32e7·24-s + 8.77e6·26-s − 1.43e7·27-s − 6.95e7·28-s + 1.78e8·29-s − 2.16e8·31-s + 1.81e8·32-s + 1.87e8·33-s + 1.48e7·34-s + ⋯ |
| L(s) = 1 | + 0.660·2-s − 0.577·3-s − 0.564·4-s − 0.381·6-s + 1.35·7-s − 1.03·8-s + 0.333·9-s − 1.44·11-s + 0.325·12-s + 0.219·13-s + 0.893·14-s − 0.117·16-s + 0.0848·17-s + 0.220·18-s − 1.30·19-s − 0.781·21-s − 0.952·22-s − 0.170·23-s + 0.596·24-s + 0.144·26-s − 0.192·27-s − 0.763·28-s + 1.61·29-s − 1.35·31-s + 0.954·32-s + 0.832·33-s + 0.0560·34-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
| \(L(6)\) |
\(\approx\) |
\(1.771670373\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.771670373\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 - 29.8T + 2.04e3T^{2} \) |
| 7 | \( 1 - 6.01e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 7.70e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 2.93e5T + 1.79e12T^{2} \) |
| 17 | \( 1 - 4.97e5T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.40e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 5.26e6T + 9.52e14T^{2} \) |
| 29 | \( 1 - 1.78e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 2.16e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 5.23e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 7.77e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.14e9T + 9.29e17T^{2} \) |
| 47 | \( 1 - 2.31e8T + 2.47e18T^{2} \) |
| 53 | \( 1 - 2.44e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 3.13e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 1.03e10T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.56e10T + 1.22e20T^{2} \) |
| 71 | \( 1 - 2.49e8T + 2.31e20T^{2} \) |
| 73 | \( 1 + 1.64e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 3.02e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 1.86e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 8.20e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 7.91e10T + 7.15e21T^{2} \) |
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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
−12.44313377851900413874241198489, −11.22983449471076011751680954298, −10.33867491504421415659552602299, −8.721998570217715619608792946391, −7.73498407435062557038314749540, −5.96598829892072457143370950505, −5.02259240664414017448870264265, −4.20653268208046993148959506026, −2.39185091644328229119199953264, −0.67396815517567529183748950542,
0.67396815517567529183748950542, 2.39185091644328229119199953264, 4.20653268208046993148959506026, 5.02259240664414017448870264265, 5.96598829892072457143370950505, 7.73498407435062557038314749540, 8.721998570217715619608792946391, 10.33867491504421415659552602299, 11.22983449471076011751680954298, 12.44313377851900413874241198489
