| L(s) = 1 | − 4·2-s − 9·3-s + 16·4-s + 46·5-s + 36·6-s + 2·7-s − 64·8-s + 81·9-s − 184·10-s + 142·11-s − 144·12-s − 8·14-s − 414·15-s + 256·16-s − 324·18-s + 736·20-s − 18·21-s − 568·22-s + 576·24-s + 1.49e3·25-s − 729·27-s + 32·28-s − 818·29-s + 1.65e3·30-s − 478·31-s − 1.02e3·32-s − 1.27e3·33-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 4-s + 1.83·5-s + 6-s + 2/49·7-s − 8-s + 9-s − 1.83·10-s + 1.17·11-s − 12-s − 0.0408·14-s − 1.83·15-s + 16-s − 18-s + 1.83·20-s − 0.0408·21-s − 1.17·22-s + 24-s + 2.38·25-s − 27-s + 2/49·28-s − 0.972·29-s + 1.83·30-s − 0.497·31-s − 32-s − 1.17·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.8827567307\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8827567307\) |
| \(L(3)\) |
|
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 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 - 46 T + p^{4} T^{2} \) |
| 7 | \( 1 - 2 T + p^{4} T^{2} \) |
| 11 | \( 1 - 142 T + p^{4} T^{2} \) |
| 13 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 17 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 19 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 23 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 29 | \( 1 + 818 T + p^{4} T^{2} \) |
| 31 | \( 1 + 478 T + p^{4} T^{2} \) |
| 37 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 41 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 43 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 47 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 53 | \( 1 + 3218 T + p^{4} T^{2} \) |
| 59 | \( 1 - 6862 T + p^{4} T^{2} \) |
| 61 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 67 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 71 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 73 | \( 1 + 8158 T + p^{4} T^{2} \) |
| 79 | \( 1 + 9118 T + p^{4} T^{2} \) |
| 83 | \( 1 + 4178 T + p^{4} T^{2} \) |
| 89 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 97 | \( 1 - 17282 T + p^{4} T^{2} \) |
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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
−17.23462374497686496412822602817, −16.37076693822842041619892272724, −14.55884187986360407340850098046, −12.89115540879677956454406529688, −11.38607733708117131672672507798, −10.11867215193048320045464497558, −9.214256717665887241136161677189, −6.78922788623161786440420384544, −5.73596391880717065078360937886, −1.57418246442969385253217428409,
1.57418246442969385253217428409, 5.73596391880717065078360937886, 6.78922788623161786440420384544, 9.214256717665887241136161677189, 10.11867215193048320045464497558, 11.38607733708117131672672507798, 12.89115540879677956454406529688, 14.55884187986360407340850098046, 16.37076693822842041619892272724, 17.23462374497686496412822602817
