| L(s) = 1 | + 4·2-s + 16·4-s − 96·5-s + 64·8-s − 384·10-s − 384·11-s + 334·13-s + 256·16-s + 576·17-s + 664·19-s − 1.53e3·20-s − 1.53e3·22-s + 3.84e3·23-s + 6.09e3·25-s + 1.33e3·26-s − 96·29-s + 4.56e3·31-s + 1.02e3·32-s + 2.30e3·34-s + 5.79e3·37-s + 2.65e3·38-s − 6.14e3·40-s − 6.72e3·41-s − 1.48e4·43-s − 6.14e3·44-s + 1.53e4·46-s − 1.92e4·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.71·5-s + 0.353·8-s − 1.21·10-s − 0.956·11-s + 0.548·13-s + 1/4·16-s + 0.483·17-s + 0.421·19-s − 0.858·20-s − 0.676·22-s + 1.51·23-s + 1.94·25-s + 0.387·26-s − 0.0211·29-s + 0.852·31-s + 0.176·32-s + 0.341·34-s + 0.696·37-s + 0.298·38-s − 0.607·40-s − 0.624·41-s − 1.22·43-s − 0.478·44-s + 1.07·46-s − 1.26·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 96 T + p^{5} T^{2} \) |
| 11 | \( 1 + 384 T + p^{5} T^{2} \) |
| 13 | \( 1 - 334 T + p^{5} T^{2} \) |
| 17 | \( 1 - 576 T + p^{5} T^{2} \) |
| 19 | \( 1 - 664 T + p^{5} T^{2} \) |
| 23 | \( 1 - 3840 T + p^{5} T^{2} \) |
| 29 | \( 1 + 96 T + p^{5} T^{2} \) |
| 31 | \( 1 - 4564 T + p^{5} T^{2} \) |
| 37 | \( 1 - 5798 T + p^{5} T^{2} \) |
| 41 | \( 1 + 6720 T + p^{5} T^{2} \) |
| 43 | \( 1 + 14872 T + p^{5} T^{2} \) |
| 47 | \( 1 + 19200 T + p^{5} T^{2} \) |
| 53 | \( 1 + 7776 T + p^{5} T^{2} \) |
| 59 | \( 1 + 13056 T + p^{5} T^{2} \) |
| 61 | \( 1 + 42782 T + p^{5} T^{2} \) |
| 67 | \( 1 - 36656 T + p^{5} T^{2} \) |
| 71 | \( 1 + 64512 T + p^{5} T^{2} \) |
| 73 | \( 1 - 16810 T + p^{5} T^{2} \) |
| 79 | \( 1 - 28076 T + p^{5} T^{2} \) |
| 83 | \( 1 + 66432 T + p^{5} T^{2} \) |
| 89 | \( 1 + 81792 T + p^{5} T^{2} \) |
| 97 | \( 1 - 29938 T + p^{5} T^{2} \) |
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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
−8.682408727925434088767409747533, −7.951782733561434016493236060749, −7.35068231695322197593529857359, −6.42159455145823920994689895654, −5.15825172929120740678693146501, −4.54749911956352791055823615966, −3.43842738261104736250776184026, −2.93225276746242718120716853192, −1.18582858672010760572552946669, 0,
1.18582858672010760572552946669, 2.93225276746242718120716853192, 3.43842738261104736250776184026, 4.54749911956352791055823615966, 5.15825172929120740678693146501, 6.42159455145823920994689895654, 7.35068231695322197593529857359, 7.951782733561434016493236060749, 8.682408727925434088767409747533
