| L(s) = 1 | + 9·3-s − 34·5-s + 240·7-s + 81·9-s + 124·11-s + 46·13-s − 306·15-s + 1.95e3·17-s + 1.92e3·19-s + 2.16e3·21-s − 2.84e3·23-s − 1.96e3·25-s + 729·27-s − 8.92e3·29-s + 4.64e3·31-s + 1.11e3·33-s − 8.16e3·35-s − 4.36e3·37-s + 414·39-s − 2.88e3·41-s − 1.13e4·43-s − 2.75e3·45-s − 7.00e3·47-s + 4.07e4·49-s + 1.75e4·51-s − 2.25e4·53-s − 4.21e3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.608·5-s + 1.85·7-s + 1/3·9-s + 0.308·11-s + 0.0754·13-s − 0.351·15-s + 1.63·17-s + 1.22·19-s + 1.06·21-s − 1.11·23-s − 0.630·25-s + 0.192·27-s − 1.97·29-s + 0.868·31-s + 0.178·33-s − 1.12·35-s − 0.523·37-s + 0.0435·39-s − 0.268·41-s − 0.934·43-s − 0.202·45-s − 0.462·47-s + 2.42·49-s + 0.946·51-s − 1.10·53-s − 0.187·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.175959555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.175959555\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 + 34 T + p^{5} T^{2} \) |
| 7 | \( 1 - 240 T + p^{5} T^{2} \) |
| 11 | \( 1 - 124 T + p^{5} T^{2} \) |
| 13 | \( 1 - 46 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1954 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1924 T + p^{5} T^{2} \) |
| 23 | \( 1 + 2840 T + p^{5} T^{2} \) |
| 29 | \( 1 + 8922 T + p^{5} T^{2} \) |
| 31 | \( 1 - 4648 T + p^{5} T^{2} \) |
| 37 | \( 1 + 4362 T + p^{5} T^{2} \) |
| 41 | \( 1 + 2886 T + p^{5} T^{2} \) |
| 43 | \( 1 + 11332 T + p^{5} T^{2} \) |
| 47 | \( 1 + 7008 T + p^{5} T^{2} \) |
| 53 | \( 1 + 22594 T + p^{5} T^{2} \) |
| 59 | \( 1 - 28 T + p^{5} T^{2} \) |
| 61 | \( 1 + 6386 T + p^{5} T^{2} \) |
| 67 | \( 1 - 39076 T + p^{5} T^{2} \) |
| 71 | \( 1 - 54872 T + p^{5} T^{2} \) |
| 73 | \( 1 - 21034 T + p^{5} T^{2} \) |
| 79 | \( 1 + 26632 T + p^{5} T^{2} \) |
| 83 | \( 1 + 56188 T + p^{5} T^{2} \) |
| 89 | \( 1 - 64410 T + p^{5} T^{2} \) |
| 97 | \( 1 + 116158 T + p^{5} T^{2} \) |
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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
−14.54495310215829336628658788607, −13.88705498674373056078845297455, −12.08185302442985684692166905811, −11.33692561187651952841388712118, −9.771206923182866300125080985498, −8.161330354653796969601598291662, −7.62699908196095682602092147308, −5.29044580880952122063261454826, −3.74871286776989221420370893437, −1.55073600233821575421852492416,
1.55073600233821575421852492416, 3.74871286776989221420370893437, 5.29044580880952122063261454826, 7.62699908196095682602092147308, 8.161330354653796969601598291662, 9.771206923182866300125080985498, 11.33692561187651952841388712118, 12.08185302442985684692166905811, 13.88705498674373056078845297455, 14.54495310215829336628658788607
