| L(s) = 1 | − 4.64·2-s + 22.5·3-s − 10.4·4-s + 9.65·5-s − 104.·6-s + 199.·7-s + 197.·8-s + 266.·9-s − 44.8·10-s + 201.·11-s − 235.·12-s − 120.·13-s − 924.·14-s + 217.·15-s − 580.·16-s − 2.03e3·17-s − 1.23e3·18-s + 724.·19-s − 100.·20-s + 4.49e3·21-s − 934.·22-s + 529·23-s + 4.44e3·24-s − 3.03e3·25-s + 557.·26-s + 530.·27-s − 2.07e3·28-s + ⋯ |
| L(s) = 1 | − 0.820·2-s + 1.44·3-s − 0.326·4-s + 0.172·5-s − 1.18·6-s + 1.53·7-s + 1.08·8-s + 1.09·9-s − 0.141·10-s + 0.501·11-s − 0.472·12-s − 0.197·13-s − 1.26·14-s + 0.250·15-s − 0.567·16-s − 1.71·17-s − 0.900·18-s + 0.460·19-s − 0.0563·20-s + 2.22·21-s − 0.411·22-s + 0.208·23-s + 1.57·24-s − 0.970·25-s + 0.161·26-s + 0.140·27-s − 0.501·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.494322098\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.494322098\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - 529T \) |
| good | 2 | \( 1 + 4.64T + 32T^{2} \) |
| 3 | \( 1 - 22.5T + 243T^{2} \) |
| 5 | \( 1 - 9.65T + 3.12e3T^{2} \) |
| 7 | \( 1 - 199.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 201.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 120.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.03e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 724.T + 2.47e6T^{2} \) |
| 29 | \( 1 + 6.36e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.29e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.04e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.18e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.48e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.35e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.24e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.90e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.33e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.95e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.31e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.19e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.88e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.88e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.35e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 7.39e4T + 8.58e9T^{2} \) |
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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
−17.25815493151074219502696955015, −15.31925096947368115050985522020, −14.22270747444329530911034470305, −13.44951431711297523666537118990, −11.21711005377386065063895384566, −9.520234629305712375355815114284, −8.582449429557738459416971039156, −7.64345664114657756060347485032, −4.42645004611278936961863670594, −1.84889158008694592119018416212,
1.84889158008694592119018416212, 4.42645004611278936961863670594, 7.64345664114657756060347485032, 8.582449429557738459416971039156, 9.520234629305712375355815114284, 11.21711005377386065063895384566, 13.44951431711297523666537118990, 14.22270747444329530911034470305, 15.31925096947368115050985522020, 17.25815493151074219502696955015
