| L(s) = 1 | − 4·2-s + 16·4-s − 21·5-s − 143·7-s − 64·8-s + 84·10-s + 205·11-s − 78·13-s + 572·14-s + 256·16-s + 2.12e3·17-s + 361·19-s − 336·20-s − 820·22-s − 20·23-s − 2.68e3·25-s + 312·26-s − 2.28e3·28-s + 4.86e3·29-s − 1.09e3·31-s − 1.02e3·32-s − 8.50e3·34-s + 3.00e3·35-s − 1.51e4·37-s − 1.44e3·38-s + 1.34e3·40-s + 9.40e3·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.375·5-s − 1.10·7-s − 0.353·8-s + 0.265·10-s + 0.510·11-s − 0.128·13-s + 0.779·14-s + 1/4·16-s + 1.78·17-s + 0.229·19-s − 0.187·20-s − 0.361·22-s − 0.00788·23-s − 0.858·25-s + 0.0905·26-s − 0.551·28-s + 1.07·29-s − 0.205·31-s − 0.176·32-s − 1.26·34-s + 0.414·35-s − 1.81·37-s − 0.162·38-s + 0.132·40-s + 0.873·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 + 21 T + p^{5} T^{2} \) |
| 7 | \( 1 + 143 T + p^{5} T^{2} \) |
| 11 | \( 1 - 205 T + p^{5} T^{2} \) |
| 13 | \( 1 + 6 p T + p^{5} T^{2} \) |
| 17 | \( 1 - 125 p T + p^{5} T^{2} \) |
| 23 | \( 1 + 20 T + p^{5} T^{2} \) |
| 29 | \( 1 - 4866 T + p^{5} T^{2} \) |
| 31 | \( 1 + 1098 T + p^{5} T^{2} \) |
| 37 | \( 1 + 15128 T + p^{5} T^{2} \) |
| 41 | \( 1 - 9400 T + p^{5} T^{2} \) |
| 43 | \( 1 - 20073 T + p^{5} T^{2} \) |
| 47 | \( 1 + 14105 T + p^{5} T^{2} \) |
| 53 | \( 1 + 26386 T + p^{5} T^{2} \) |
| 59 | \( 1 - 224 p T + p^{5} T^{2} \) |
| 61 | \( 1 + 2293 T + p^{5} T^{2} \) |
| 67 | \( 1 - 35976 T + p^{5} T^{2} \) |
| 71 | \( 1 + 10180 T + p^{5} T^{2} \) |
| 73 | \( 1 - 33109 T + p^{5} T^{2} \) |
| 79 | \( 1 + 53888 T + p^{5} T^{2} \) |
| 83 | \( 1 + 75196 T + p^{5} T^{2} \) |
| 89 | \( 1 + 20618 T + p^{5} T^{2} \) |
| 97 | \( 1 + 84130 T + p^{5} T^{2} \) |
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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
−9.952509102990119204041015773477, −9.516686368245846512422018450307, −8.352078725225545321150375494887, −7.45452708764762014297803808520, −6.53186012806828306900359934904, −5.51944299219090269981811399898, −3.83671519706570632670995559198, −2.90990564895872417536443266305, −1.24948842169905804378590004665, 0,
1.24948842169905804378590004665, 2.90990564895872417536443266305, 3.83671519706570632670995559198, 5.51944299219090269981811399898, 6.53186012806828306900359934904, 7.45452708764762014297803808520, 8.352078725225545321150375494887, 9.516686368245846512422018450307, 9.952509102990119204041015773477
