| L(s) = 1 | + 140·2-s + 729·3-s + 1.14e4·4-s + 4.87e4·5-s + 1.02e5·6-s + 4.87e5·7-s + 4.50e5·8-s + 5.31e5·9-s + 6.82e6·10-s − 1.77e6·11-s + 8.31e6·12-s − 1.83e7·13-s + 6.82e7·14-s + 3.55e7·15-s − 3.04e7·16-s − 9.62e7·17-s + 7.44e7·18-s − 1.49e7·19-s + 5.56e8·20-s + 3.55e8·21-s − 2.48e8·22-s + 1.53e8·23-s + 3.28e8·24-s + 1.15e9·25-s − 2.57e9·26-s + 3.87e8·27-s + 5.56e9·28-s + ⋯ |
| L(s) = 1 | + 1.54·2-s + 0.577·3-s + 1.39·4-s + 1.39·5-s + 0.893·6-s + 1.56·7-s + 0.607·8-s + 1/3·9-s + 2.15·10-s − 0.301·11-s + 0.804·12-s − 1.05·13-s + 2.42·14-s + 0.805·15-s − 0.453·16-s − 0.966·17-s + 0.515·18-s − 0.0729·19-s + 1.94·20-s + 0.904·21-s − 0.466·22-s + 0.216·23-s + 0.350·24-s + 0.946·25-s − 1.63·26-s + 0.192·27-s + 2.18·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(7.891163092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.891163092\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p^{6} T \) |
| 11 | \( 1 + p^{6} T \) |
| good | 2 | \( 1 - 35 p^{2} T + p^{13} T^{2} \) |
| 5 | \( 1 - 9748 p T + p^{13} T^{2} \) |
| 7 | \( 1 - 487486 T + p^{13} T^{2} \) |
| 13 | \( 1 + 18388304 T + p^{13} T^{2} \) |
| 17 | \( 1 + 96233254 T + p^{13} T^{2} \) |
| 19 | \( 1 + 14954652 T + p^{13} T^{2} \) |
| 23 | \( 1 - 153804394 T + p^{13} T^{2} \) |
| 29 | \( 1 - 5219010534 T + p^{13} T^{2} \) |
| 31 | \( 1 - 1183811728 T + p^{13} T^{2} \) |
| 37 | \( 1 + 17672200362 T + p^{13} T^{2} \) |
| 41 | \( 1 + 19461739306 T + p^{13} T^{2} \) |
| 43 | \( 1 + 79355928 p T + p^{13} T^{2} \) |
| 47 | \( 1 + 100327719050 T + p^{13} T^{2} \) |
| 53 | \( 1 - 275469097716 T + p^{13} T^{2} \) |
| 59 | \( 1 + 267676863080 T + p^{13} T^{2} \) |
| 61 | \( 1 - 563486626260 T + p^{13} T^{2} \) |
| 67 | \( 1 - 1080842815700 T + p^{13} T^{2} \) |
| 71 | \( 1 + 1150562265222 T + p^{13} T^{2} \) |
| 73 | \( 1 + 345914515454 T + p^{13} T^{2} \) |
| 79 | \( 1 + 2004080959294 T + p^{13} T^{2} \) |
| 83 | \( 1 + 3336732240564 T + p^{13} T^{2} \) |
| 89 | \( 1 + 5696238036294 T + p^{13} T^{2} \) |
| 97 | \( 1 + 6550114593202 T + p^{13} T^{2} \) |
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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
−13.94300358636676800050025533610, −12.98653630925601026572263765168, −11.67272570689685354473724827356, −10.21821014168884657375674165636, −8.599698246917419019228463224904, −6.84789568417933628640424568079, −5.32476514760438560927958477871, −4.57250189909968270950748693320, −2.63112862577717729411360416363, −1.80544993865997630116203589564,
1.80544993865997630116203589564, 2.63112862577717729411360416363, 4.57250189909968270950748693320, 5.32476514760438560927958477871, 6.84789568417933628640424568079, 8.599698246917419019228463224904, 10.21821014168884657375674165636, 11.67272570689685354473724827356, 12.98653630925601026572263765168, 13.94300358636676800050025533610
