| L(s) = 1 | − 2·2-s − 9·3-s − 28·4-s − 25·5-s + 18·6-s − 132·7-s + 120·8-s + 81·9-s + 50·10-s + 472·11-s + 252·12-s − 686·13-s + 264·14-s + 225·15-s + 656·16-s − 1.56e3·17-s − 162·18-s − 2.18e3·19-s + 700·20-s + 1.18e3·21-s − 944·22-s + 264·23-s − 1.08e3·24-s + 625·25-s + 1.37e3·26-s − 729·27-s + 3.69e3·28-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 0.577·3-s − 7/8·4-s − 0.447·5-s + 0.204·6-s − 1.01·7-s + 0.662·8-s + 1/3·9-s + 0.158·10-s + 1.17·11-s + 0.505·12-s − 1.12·13-s + 0.359·14-s + 0.258·15-s + 0.640·16-s − 1.31·17-s − 0.117·18-s − 1.38·19-s + 0.391·20-s + 0.587·21-s − 0.415·22-s + 0.104·23-s − 0.382·24-s + 1/5·25-s + 0.398·26-s − 0.192·27-s + 0.890·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p^{2} T \) |
| 5 | \( 1 + p^{2} T \) |
| good | 2 | \( 1 + p T + p^{5} T^{2} \) |
| 7 | \( 1 + 132 T + p^{5} T^{2} \) |
| 11 | \( 1 - 472 T + p^{5} T^{2} \) |
| 13 | \( 1 + 686 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1562 T + p^{5} T^{2} \) |
| 19 | \( 1 + 2180 T + p^{5} T^{2} \) |
| 23 | \( 1 - 264 T + p^{5} T^{2} \) |
| 29 | \( 1 - 170 T + p^{5} T^{2} \) |
| 31 | \( 1 - 7272 T + p^{5} T^{2} \) |
| 37 | \( 1 + 142 T + p^{5} T^{2} \) |
| 41 | \( 1 + 16198 T + p^{5} T^{2} \) |
| 43 | \( 1 + 10316 T + p^{5} T^{2} \) |
| 47 | \( 1 - 18568 T + p^{5} T^{2} \) |
| 53 | \( 1 - 21514 T + p^{5} T^{2} \) |
| 59 | \( 1 - 34600 T + p^{5} T^{2} \) |
| 61 | \( 1 + 35738 T + p^{5} T^{2} \) |
| 67 | \( 1 + 5772 T + p^{5} T^{2} \) |
| 71 | \( 1 + 69088 T + p^{5} T^{2} \) |
| 73 | \( 1 + 70526 T + p^{5} T^{2} \) |
| 79 | \( 1 - 47640 T + p^{5} T^{2} \) |
| 83 | \( 1 - 74004 T + p^{5} T^{2} \) |
| 89 | \( 1 + 90030 T + p^{5} T^{2} \) |
| 97 | \( 1 + 33502 T + p^{5} T^{2} \) |
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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
−17.48841249195048666411803783111, −16.70804256084611237851489891586, −15.05725837507799007067983190145, −13.32379307699233665460796833533, −12.04621202083319274507530947614, −10.20194959454477331101819983496, −8.854131747055328688060587257470, −6.71144390140536460468990038079, −4.35806742067215223865182507911, 0,
4.35806742067215223865182507911, 6.71144390140536460468990038079, 8.854131747055328688060587257470, 10.20194959454477331101819983496, 12.04621202083319274507530947614, 13.32379307699233665460796833533, 15.05725837507799007067983190145, 16.70804256084611237851489891586, 17.48841249195048666411803783111
