| L(s) = 1 | − 8·2-s − 36.0·3-s + 64·4-s − 19.7·5-s + 288.·6-s − 50.9·7-s − 512·8-s − 886.·9-s + 157.·10-s + 5.17e3·11-s − 2.30e3·12-s + 4.31e3·13-s + 407.·14-s + 711.·15-s + 4.09e3·16-s + 1.40e4·17-s + 7.09e3·18-s + 2.66e4·19-s − 1.26e3·20-s + 1.83e3·21-s − 4.14e4·22-s − 7.74e4·23-s + 1.84e4·24-s − 7.77e4·25-s − 3.44e4·26-s + 1.10e5·27-s − 3.25e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.771·3-s + 0.5·4-s − 0.0706·5-s + 0.545·6-s − 0.0561·7-s − 0.353·8-s − 0.405·9-s + 0.0499·10-s + 1.17·11-s − 0.385·12-s + 0.544·13-s + 0.0396·14-s + 0.0544·15-s + 0.250·16-s + 0.694·17-s + 0.286·18-s + 0.891·19-s − 0.0353·20-s + 0.0432·21-s − 0.829·22-s − 1.32·23-s + 0.272·24-s − 0.995·25-s − 0.384·26-s + 1.08·27-s − 0.0280·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 37 | \( 1 - 5.06e4T \) |
| good | 3 | \( 1 + 36.0T + 2.18e3T^{2} \) |
| 5 | \( 1 + 19.7T + 7.81e4T^{2} \) |
| 7 | \( 1 + 50.9T + 8.23e5T^{2} \) |
| 11 | \( 1 - 5.17e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 4.31e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.40e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.66e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 7.74e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 3.31e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.72e5T + 2.75e10T^{2} \) |
| 41 | \( 1 + 8.46e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.07e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 3.22e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.83e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 7.97e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 6.84e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.79e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.42e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.64e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.44e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.61e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 8.43e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 3.40e6T + 8.07e13T^{2} \) |
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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
−11.97516636307086132533139138572, −11.61792304272230807525287413550, −10.32215851552829083112350510814, −9.221640083167962114200095471210, −7.975799993867005169480289000987, −6.56161858019835799145226513004, −5.55486254918469052860634700054, −3.58610015018173937621774895127, −1.48555689459506694529918385714, 0,
1.48555689459506694529918385714, 3.58610015018173937621774895127, 5.55486254918469052860634700054, 6.56161858019835799145226513004, 7.975799993867005169480289000987, 9.221640083167962114200095471210, 10.32215851552829083112350510814, 11.61792304272230807525287413550, 11.97516636307086132533139138572
