| L(s) = 1 | + 8·2-s + 64·4-s + 120·5-s + 377·7-s + 512·8-s + 960·10-s + 600·11-s + 5.36e3·13-s + 3.01e3·14-s + 4.09e3·16-s + 1.21e4·17-s + 1.62e4·19-s + 7.68e3·20-s + 4.80e3·22-s + 1.06e5·23-s − 6.37e4·25-s + 4.29e4·26-s + 2.41e4·28-s + 1.77e5·29-s − 2.68e5·31-s + 3.27e4·32-s + 9.73e4·34-s + 4.52e4·35-s + 1.14e5·37-s + 1.29e5·38-s + 6.14e4·40-s − 1.12e5·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.429·5-s + 0.415·7-s + 0.353·8-s + 0.303·10-s + 0.135·11-s + 0.677·13-s + 0.293·14-s + 1/4·16-s + 0.600·17-s + 0.542·19-s + 0.214·20-s + 0.0961·22-s + 1.82·23-s − 0.815·25-s + 0.479·26-s + 0.207·28-s + 1.34·29-s − 1.61·31-s + 0.176·32-s + 0.424·34-s + 0.178·35-s + 0.373·37-s + 0.383·38-s + 0.151·40-s − 0.254·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.385782465\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.385782465\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{3} T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 24 p T + p^{7} T^{2} \) |
| 7 | \( 1 - 377 T + p^{7} T^{2} \) |
| 11 | \( 1 - 600 T + p^{7} T^{2} \) |
| 13 | \( 1 - 413 p T + p^{7} T^{2} \) |
| 17 | \( 1 - 12168 T + p^{7} T^{2} \) |
| 19 | \( 1 - 16211 T + p^{7} T^{2} \) |
| 23 | \( 1 - 106392 T + p^{7} T^{2} \) |
| 29 | \( 1 - 177216 T + p^{7} T^{2} \) |
| 31 | \( 1 + 268060 T + p^{7} T^{2} \) |
| 37 | \( 1 - 3107 p T + p^{7} T^{2} \) |
| 41 | \( 1 + 112128 T + p^{7} T^{2} \) |
| 43 | \( 1 + 115048 T + p^{7} T^{2} \) |
| 47 | \( 1 - 561336 T + p^{7} T^{2} \) |
| 53 | \( 1 + 1787760 T + p^{7} T^{2} \) |
| 59 | \( 1 + 1786344 T + p^{7} T^{2} \) |
| 61 | \( 1 + 1306837 T + p^{7} T^{2} \) |
| 67 | \( 1 + 2013817 T + p^{7} T^{2} \) |
| 71 | \( 1 + 4060944 T + p^{7} T^{2} \) |
| 73 | \( 1 + 3850639 T + p^{7} T^{2} \) |
| 79 | \( 1 - 1037231 T + p^{7} T^{2} \) |
| 83 | \( 1 - 9203568 T + p^{7} T^{2} \) |
| 89 | \( 1 + 1289304 T + p^{7} T^{2} \) |
| 97 | \( 1 - 88205 p T + p^{7} T^{2} \) |
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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
−13.86867928058926597482648858993, −12.88681696965334178388850609696, −11.65172943096396841762973300617, −10.59619386209118130026424453878, −9.121085939868488568377273578417, −7.56692515506943755919618262673, −6.12058908911325826341898168948, −4.88279479120442117985688092904, −3.22013881846311690052507274075, −1.42073146658780787921852952739,
1.42073146658780787921852952739, 3.22013881846311690052507274075, 4.88279479120442117985688092904, 6.12058908911325826341898168948, 7.56692515506943755919618262673, 9.121085939868488568377273578417, 10.59619386209118130026424453878, 11.65172943096396841762973300617, 12.88681696965334178388850609696, 13.86867928058926597482648858993
