| L(s) = 1 | − 9·3-s − 54·5-s − 104·7-s + 81·9-s + 330·11-s − 46·13-s + 486·15-s − 618·17-s − 361·19-s + 936·21-s + 402·23-s − 209·25-s − 729·27-s − 2.62e3·29-s + 2.36e3·31-s − 2.97e3·33-s + 5.61e3·35-s − 1.21e4·37-s + 414·39-s − 1.88e4·41-s + 1.04e4·43-s − 4.37e3·45-s + 4.77e3·47-s − 5.99e3·49-s + 5.56e3·51-s − 1.94e4·53-s − 1.78e4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.965·5-s − 0.802·7-s + 1/3·9-s + 0.822·11-s − 0.0754·13-s + 0.557·15-s − 0.518·17-s − 0.229·19-s + 0.463·21-s + 0.158·23-s − 0.0668·25-s − 0.192·27-s − 0.580·29-s + 0.442·31-s − 0.474·33-s + 0.774·35-s − 1.45·37-s + 0.0435·39-s − 1.75·41-s + 0.858·43-s − 0.321·45-s + 0.314·47-s − 0.356·49-s + 0.299·51-s − 0.951·53-s − 0.794·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.5279299315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5279299315\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{2} T \) |
| 19 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 + 54 T + p^{5} T^{2} \) |
| 7 | \( 1 + 104 T + p^{5} T^{2} \) |
| 11 | \( 1 - 30 p T + p^{5} T^{2} \) |
| 13 | \( 1 + 46 T + p^{5} T^{2} \) |
| 17 | \( 1 + 618 T + p^{5} T^{2} \) |
| 23 | \( 1 - 402 T + p^{5} T^{2} \) |
| 29 | \( 1 + 2628 T + p^{5} T^{2} \) |
| 31 | \( 1 - 2368 T + p^{5} T^{2} \) |
| 37 | \( 1 + 12130 T + p^{5} T^{2} \) |
| 41 | \( 1 + 18864 T + p^{5} T^{2} \) |
| 43 | \( 1 - 10408 T + p^{5} T^{2} \) |
| 47 | \( 1 - 4770 T + p^{5} T^{2} \) |
| 53 | \( 1 + 19452 T + p^{5} T^{2} \) |
| 59 | \( 1 + 30528 T + p^{5} T^{2} \) |
| 61 | \( 1 - 11138 T + p^{5} T^{2} \) |
| 67 | \( 1 + 49508 T + p^{5} T^{2} \) |
| 71 | \( 1 + 7572 T + p^{5} T^{2} \) |
| 73 | \( 1 - 2342 T + p^{5} T^{2} \) |
| 79 | \( 1 + 22424 T + p^{5} T^{2} \) |
| 83 | \( 1 - 46734 T + p^{5} T^{2} \) |
| 89 | \( 1 + 70104 T + p^{5} T^{2} \) |
| 97 | \( 1 - 105710 T + p^{5} T^{2} \) |
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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
−9.344609162207410367768075489974, −8.567015209724859307575485216278, −7.50771335869958083661693555558, −6.76230324248476894147518426848, −6.06362638032569195607839007589, −4.86992569219605215559856718215, −3.97697538226408777281271315033, −3.20294688703784200765036653309, −1.67268856680033577133971851547, −0.33116197167016676691995846755,
0.33116197167016676691995846755, 1.67268856680033577133971851547, 3.20294688703784200765036653309, 3.97697538226408777281271315033, 4.86992569219605215559856718215, 6.06362638032569195607839007589, 6.76230324248476894147518426848, 7.50771335869958083661693555558, 8.567015209724859307575485216278, 9.344609162207410367768075489974
