| L(s) = 1 | − 2.04e3·2-s + 1.77e5·3-s + 4.19e6·4-s − 3.54e7·5-s − 3.62e8·6-s − 2.38e9·7-s − 8.58e9·8-s + 3.13e10·9-s + 7.26e10·10-s + 4.27e11·11-s + 7.43e11·12-s + 4.30e12·13-s + 4.88e12·14-s − 6.28e12·15-s + 1.75e13·16-s − 2.11e14·17-s − 6.42e13·18-s − 3.03e14·19-s − 1.48e14·20-s − 4.22e14·21-s − 8.76e14·22-s − 4.08e15·23-s − 1.52e15·24-s − 1.06e16·25-s − 8.81e15·26-s + 5.55e15·27-s − 1.00e16·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.324·5-s − 0.408·6-s − 0.456·7-s − 0.353·8-s + 1/3·9-s + 0.229·10-s + 0.452·11-s + 0.288·12-s + 0.666·13-s + 0.322·14-s − 0.187·15-s + 1/4·16-s − 1.49·17-s − 0.235·18-s − 0.597·19-s − 0.162·20-s − 0.263·21-s − 0.319·22-s − 0.893·23-s − 0.204·24-s − 0.894·25-s − 0.470·26-s + 0.192·27-s − 0.228·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{25}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{11} T \) |
| 3 | \( 1 - p^{11} T \) |
| good | 5 | \( 1 + 283866 p^{3} T + p^{23} T^{2} \) |
| 7 | \( 1 + 340835416 p T + p^{23} T^{2} \) |
| 11 | \( 1 - 38894122860 p T + p^{23} T^{2} \) |
| 13 | \( 1 - 4303510800614 T + p^{23} T^{2} \) |
| 17 | \( 1 + 12445098770286 p T + p^{23} T^{2} \) |
| 19 | \( 1 + 303299666491876 T + p^{23} T^{2} \) |
| 23 | \( 1 + 4084826356392600 T + p^{23} T^{2} \) |
| 29 | \( 1 + 76724512266210954 T + p^{23} T^{2} \) |
| 31 | \( 1 + 95662499637633472 T + p^{23} T^{2} \) |
| 37 | \( 1 - 1916787087325361486 T + p^{23} T^{2} \) |
| 41 | \( 1 + 3821928337631245926 T + p^{23} T^{2} \) |
| 43 | \( 1 + 5028833488465187068 T + p^{23} T^{2} \) |
| 47 | \( 1 + 20587597004644658160 T + p^{23} T^{2} \) |
| 53 | \( 1 + 17205347518114927842 T + p^{23} T^{2} \) |
| 59 | \( 1 - \)\(10\!\cdots\!40\)\( T + p^{23} T^{2} \) |
| 61 | \( 1 - \)\(47\!\cdots\!70\)\( T + p^{23} T^{2} \) |
| 67 | \( 1 - \)\(47\!\cdots\!48\)\( T + p^{23} T^{2} \) |
| 71 | \( 1 + \)\(30\!\cdots\!20\)\( T + p^{23} T^{2} \) |
| 73 | \( 1 - \)\(46\!\cdots\!54\)\( T + p^{23} T^{2} \) |
| 79 | \( 1 - \)\(96\!\cdots\!12\)\( T + p^{23} T^{2} \) |
| 83 | \( 1 - \)\(90\!\cdots\!04\)\( T + p^{23} T^{2} \) |
| 89 | \( 1 + \)\(78\!\cdots\!18\)\( T + p^{23} T^{2} \) |
| 97 | \( 1 + \)\(57\!\cdots\!94\)\( T + p^{23} T^{2} \) |
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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
−16.34223698622922691324816466206, −15.06958111532552916372792702258, −13.20107537253608673252799199158, −11.29565641764000692789126289663, −9.540758607489861326724197861483, −8.217209729396845168040458800632, −6.52614872901513040323843593533, −3.80865645511090602599102217096, −1.96036747503661344468239952421, 0,
1.96036747503661344468239952421, 3.80865645511090602599102217096, 6.52614872901513040323843593533, 8.217209729396845168040458800632, 9.540758607489861326724197861483, 11.29565641764000692789126289663, 13.20107537253608673252799199158, 15.06958111532552916372792702258, 16.34223698622922691324816466206
