| L(s) = 1 | + 21.5·2-s − 27·3-s + 337.·4-s − 205.·5-s − 582.·6-s + 1.23e3·7-s + 4.53e3·8-s + 729·9-s − 4.43e3·10-s + 3.41e3·11-s − 9.12e3·12-s + 1.22e4·13-s + 2.65e4·14-s + 5.54e3·15-s + 5.45e4·16-s − 1.55e4·17-s + 1.57e4·18-s − 3.10e4·19-s − 6.94e4·20-s − 3.32e4·21-s + 7.36e4·22-s + 3.64e4·23-s − 1.22e5·24-s − 3.58e4·25-s + 2.63e5·26-s − 1.96e4·27-s + 4.16e5·28-s + ⋯ |
| L(s) = 1 | + 1.90·2-s − 0.577·3-s + 2.63·4-s − 0.735·5-s − 1.10·6-s + 1.35·7-s + 3.12·8-s + 0.333·9-s − 1.40·10-s + 0.772·11-s − 1.52·12-s + 1.54·13-s + 2.58·14-s + 0.424·15-s + 3.32·16-s − 0.765·17-s + 0.635·18-s − 1.03·19-s − 1.94·20-s − 0.783·21-s + 1.47·22-s + 0.625·23-s − 1.80·24-s − 0.459·25-s + 2.94·26-s − 0.192·27-s + 3.58·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(8.625177301\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.625177301\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 27T \) |
| 157 | \( 1 + 3.86e6T \) |
| good | 2 | \( 1 - 21.5T + 128T^{2} \) |
| 5 | \( 1 + 205.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.23e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 3.41e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.22e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.55e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.10e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.64e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 6.66e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.59e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.94e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.55e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.34e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.08e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.69e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.01e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.84e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 6.81e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.68e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.91e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 7.53e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 3.80e5T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.07e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + 4.86e6T + 8.07e13T^{2} \) |
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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
−10.73463532634141019458621447703, −8.660594669798415870612441759948, −7.73492386250300110858897155547, −6.62897062666194575935183292536, −6.05502901323501767339291638084, −4.86276710182902363715788626725, −4.31287049539532017371457740225, −3.55668117984818947682622757470, −2.05590230860884295255506350239, −1.10439121527151110032839852624,
1.10439121527151110032839852624, 2.05590230860884295255506350239, 3.55668117984818947682622757470, 4.31287049539532017371457740225, 4.86276710182902363715788626725, 6.05502901323501767339291638084, 6.62897062666194575935183292536, 7.73492386250300110858897155547, 8.660594669798415870612441759948, 10.73463532634141019458621447703
