| L(s) = 1 | + 9.60·2-s − 35.7·4-s − 26.4·5-s − 248.·7-s − 1.57e3·8-s − 253.·10-s − 43.2·11-s − 5.82e3·13-s − 2.38e3·14-s − 1.05e4·16-s − 4.64e3·17-s + 4.54e4·19-s + 944.·20-s − 415.·22-s + 1.21e4·23-s − 7.74e4·25-s − 5.59e4·26-s + 8.86e3·28-s + 2.57e5·29-s − 1.40e5·31-s + 1.00e5·32-s − 4.46e4·34-s + 6.56e3·35-s + 1.54e5·37-s + 4.36e5·38-s + 4.15e4·40-s + 6.33e5·41-s + ⋯ |
| L(s) = 1 | + 0.849·2-s − 0.278·4-s − 0.0945·5-s − 0.273·7-s − 1.08·8-s − 0.0803·10-s − 0.00980·11-s − 0.735·13-s − 0.232·14-s − 0.643·16-s − 0.229·17-s + 1.51·19-s + 0.0263·20-s − 0.00832·22-s + 0.208·23-s − 0.991·25-s − 0.624·26-s + 0.0762·28-s + 1.96·29-s − 0.845·31-s + 0.539·32-s − 0.194·34-s + 0.0258·35-s + 0.501·37-s + 1.29·38-s + 0.102·40-s + 1.43·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.196755026\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.196755026\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - 1.21e4T \) |
| good | 2 | \( 1 - 9.60T + 128T^{2} \) |
| 5 | \( 1 + 26.4T + 7.81e4T^{2} \) |
| 7 | \( 1 + 248.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 43.2T + 1.94e7T^{2} \) |
| 13 | \( 1 + 5.82e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 4.64e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.54e4T + 8.93e8T^{2} \) |
| 29 | \( 1 - 2.57e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.40e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.54e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.33e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 5.42e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 5.68e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 4.28e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.15e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.75e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.84e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.59e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.46e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.28e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 3.90e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 3.48e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.32e7T + 8.07e13T^{2} \) |
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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
−11.47419630442683754021043018999, −10.00350072969879591658733060568, −9.291683527043612948849217412706, −8.075903490750853799353815617518, −6.86821878444278765960116784241, −5.67570104940546248753440768567, −4.78906001558867018208962381433, −3.67144312454892937059388447229, −2.59043870951947697348831653639, −0.68324865343716528049663421778,
0.68324865343716528049663421778, 2.59043870951947697348831653639, 3.67144312454892937059388447229, 4.78906001558867018208962381433, 5.67570104940546248753440768567, 6.86821878444278765960116784241, 8.075903490750853799353815617518, 9.291683527043612948849217412706, 10.00350072969879591658733060568, 11.47419630442683754021043018999
