| L(s) = 1 | − 27·3-s + 270·5-s − 343·7-s + 729·9-s + 1.40e3·11-s − 2.36e3·13-s − 7.29e3·15-s + 1.83e4·17-s − 2.71e4·19-s + 9.26e3·21-s − 8.61e4·23-s − 5.22e3·25-s − 1.96e4·27-s + 2.51e5·29-s + 2.70e4·31-s − 3.79e4·33-s − 9.26e4·35-s − 4.10e5·37-s + 6.37e4·39-s − 2.46e5·41-s − 4.95e4·43-s + 1.96e5·45-s + 4.09e5·47-s + 1.17e5·49-s − 4.95e5·51-s + 1.48e6·53-s + 3.79e5·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.965·5-s − 0.377·7-s + 1/3·9-s + 0.318·11-s − 0.298·13-s − 0.557·15-s + 0.906·17-s − 0.908·19-s + 0.218·21-s − 1.47·23-s − 0.0668·25-s − 0.192·27-s + 1.91·29-s + 0.163·31-s − 0.183·33-s − 0.365·35-s − 1.33·37-s + 0.172·39-s − 0.557·41-s − 0.0949·43-s + 0.321·45-s + 0.574·47-s + 1/7·49-s − 0.523·51-s + 1.37·53-s + 0.307·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + p^{3} T \) |
| 7 | \( 1 + p^{3} T \) |
| good | 5 | \( 1 - 54 p T + p^{7} T^{2} \) |
| 11 | \( 1 - 1404 T + p^{7} T^{2} \) |
| 13 | \( 1 + 2362 T + p^{7} T^{2} \) |
| 17 | \( 1 - 18354 T + p^{7} T^{2} \) |
| 19 | \( 1 + 27164 T + p^{7} T^{2} \) |
| 23 | \( 1 + 86184 T + p^{7} T^{2} \) |
| 29 | \( 1 - 251862 T + p^{7} T^{2} \) |
| 31 | \( 1 - 27040 T + p^{7} T^{2} \) |
| 37 | \( 1 + 410290 T + p^{7} T^{2} \) |
| 41 | \( 1 + 6006 p T + p^{7} T^{2} \) |
| 43 | \( 1 + 49508 T + p^{7} T^{2} \) |
| 47 | \( 1 - 409200 T + p^{7} T^{2} \) |
| 53 | \( 1 - 1486398 T + p^{7} T^{2} \) |
| 59 | \( 1 + 427956 T + p^{7} T^{2} \) |
| 61 | \( 1 + 2048554 T + p^{7} T^{2} \) |
| 67 | \( 1 + 940748 T + p^{7} T^{2} \) |
| 71 | \( 1 + 789048 T + p^{7} T^{2} \) |
| 73 | \( 1 - 374330 T + p^{7} T^{2} \) |
| 79 | \( 1 - 4260880 T + p^{7} T^{2} \) |
| 83 | \( 1 - 8772132 T + p^{7} T^{2} \) |
| 89 | \( 1 - 2703786 T + p^{7} T^{2} \) |
| 97 | \( 1 + 13666078 T + p^{7} T^{2} \) |
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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
−10.16151521061678305689526354409, −9.122526353901612206539332304467, −7.995389423630499454076658286288, −6.68875435533087491594328079341, −6.06250831725325616707963242034, −5.11604739386900419084068811173, −3.88832068654309829811578795380, −2.45193582217351599592656334655, −1.33890941117447926402990848592, 0,
1.33890941117447926402990848592, 2.45193582217351599592656334655, 3.88832068654309829811578795380, 5.11604739386900419084068811173, 6.06250831725325616707963242034, 6.68875435533087491594328079341, 7.995389423630499454076658286288, 9.122526353901612206539332304467, 10.16151521061678305689526354409
