| L(s) = 1 | − 4·2-s + 16·4-s − 4·5-s − 146·7-s − 64·8-s + 16·10-s + 620·11-s + 169·13-s + 584·14-s + 256·16-s + 302·17-s − 2.30e3·19-s − 64·20-s − 2.48e3·22-s + 2.59e3·23-s − 3.10e3·25-s − 676·26-s − 2.33e3·28-s − 1.96e3·29-s + 5.94e3·31-s − 1.02e3·32-s − 1.20e3·34-s + 584·35-s + 9.32e3·37-s + 9.22e3·38-s + 256·40-s + 4.25e3·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.0715·5-s − 1.12·7-s − 0.353·8-s + 0.0505·10-s + 1.54·11-s + 0.277·13-s + 0.796·14-s + 1/4·16-s + 0.253·17-s − 1.46·19-s − 0.0357·20-s − 1.09·22-s + 1.02·23-s − 0.994·25-s − 0.196·26-s − 0.563·28-s − 0.433·29-s + 1.11·31-s − 0.176·32-s − 0.179·34-s + 0.0805·35-s + 1.11·37-s + 1.03·38-s + 0.0252·40-s + 0.395·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + p^{2} T \) |
| 3 | \( 1 \) |
| 13 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 + 4 T + p^{5} T^{2} \) |
| 7 | \( 1 + 146 T + p^{5} T^{2} \) |
| 11 | \( 1 - 620 T + p^{5} T^{2} \) |
| 17 | \( 1 - 302 T + p^{5} T^{2} \) |
| 19 | \( 1 + 2306 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2592 T + p^{5} T^{2} \) |
| 29 | \( 1 + 1962 T + p^{5} T^{2} \) |
| 31 | \( 1 - 5942 T + p^{5} T^{2} \) |
| 37 | \( 1 - 9322 T + p^{5} T^{2} \) |
| 41 | \( 1 - 4256 T + p^{5} T^{2} \) |
| 43 | \( 1 + 5948 T + p^{5} T^{2} \) |
| 47 | \( 1 + 17140 T + p^{5} T^{2} \) |
| 53 | \( 1 + 19750 T + p^{5} T^{2} \) |
| 59 | \( 1 + 31520 T + p^{5} T^{2} \) |
| 61 | \( 1 + 50270 T + p^{5} T^{2} \) |
| 67 | \( 1 + 26254 T + p^{5} T^{2} \) |
| 71 | \( 1 + 56744 T + p^{5} T^{2} \) |
| 73 | \( 1 - 38534 T + p^{5} T^{2} \) |
| 79 | \( 1 + 32608 T + p^{5} T^{2} \) |
| 83 | \( 1 - 116424 T + p^{5} T^{2} \) |
| 89 | \( 1 + 71236 T + p^{5} T^{2} \) |
| 97 | \( 1 + 128786 T + p^{5} T^{2} \) |
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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
−10.73539329417059822752095029194, −9.620639623537400963983116919015, −9.098511332755152460738370511341, −7.954730154094437619303559616025, −6.62090638756478769891374624932, −6.18808885777394885506443304365, −4.23466471266623295872486655879, −3.04900634200505419657424608232, −1.43793260089473531159462527155, 0,
1.43793260089473531159462527155, 3.04900634200505419657424608232, 4.23466471266623295872486655879, 6.18808885777394885506443304365, 6.62090638756478769891374624932, 7.954730154094437619303559616025, 9.098511332755152460738370511341, 9.620639623537400963983116919015, 10.73539329417059822752095029194
