| L(s) = 1 | − 31.8·2-s + 81·3-s + 504.·4-s − 1.88e3·5-s − 2.58e3·6-s + 1.22e4·7-s + 225.·8-s + 6.56e3·9-s + 6.02e4·10-s − 1.70e4·11-s + 4.08e4·12-s + 2.77e3·13-s − 3.91e5·14-s − 1.53e5·15-s − 2.65e5·16-s − 3.07e5·17-s − 2.09e5·18-s + 3.34e5·19-s − 9.53e5·20-s + 9.94e5·21-s + 5.43e5·22-s − 8.29e5·23-s + 1.82e4·24-s + 1.61e6·25-s − 8.83e4·26-s + 5.31e5·27-s + 6.19e6·28-s + ⋯ |
| L(s) = 1 | − 1.40·2-s + 0.577·3-s + 0.986·4-s − 1.35·5-s − 0.813·6-s + 1.93·7-s + 0.0194·8-s + 0.333·9-s + 1.90·10-s − 0.351·11-s + 0.569·12-s + 0.0269·13-s − 2.72·14-s − 0.780·15-s − 1.01·16-s − 0.892·17-s − 0.469·18-s + 0.589·19-s − 1.33·20-s + 1.11·21-s + 0.494·22-s − 0.618·23-s + 0.0112·24-s + 0.827·25-s − 0.0379·26-s + 0.192·27-s + 1.90·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 81T \) |
| 59 | \( 1 - 1.21e7T \) |
| good | 2 | \( 1 + 31.8T + 512T^{2} \) |
| 5 | \( 1 + 1.88e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.22e4T + 4.03e7T^{2} \) |
| 11 | \( 1 + 1.70e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 2.77e3T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.07e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 3.34e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 8.29e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.06e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 8.06e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 2.98e5T + 1.29e14T^{2} \) |
| 41 | \( 1 - 9.57e5T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.34e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.47e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.59e7T + 3.29e15T^{2} \) |
| 61 | \( 1 - 4.27e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.05e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.00e6T + 4.58e16T^{2} \) |
| 73 | \( 1 - 4.31e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.46e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.37e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 5.27e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 9.13e8T + 7.60e17T^{2} \) |
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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
−10.55477076684853104955072928587, −9.192806650460643580872783284709, −8.228652722409017783625495863881, −7.953673999707880210361994886026, −7.13206375873013511654360777662, −4.92703636913103047486797399963, −3.99796734962131603356522808092, −2.23817075882151003970999567278, −1.21602035915372520097497812932, 0,
1.21602035915372520097497812932, 2.23817075882151003970999567278, 3.99796734962131603356522808092, 4.92703636913103047486797399963, 7.13206375873013511654360777662, 7.953673999707880210361994886026, 8.228652722409017783625495863881, 9.192806650460643580872783284709, 10.55477076684853104955072928587
