| L(s) = 1 | + 512·2-s − 1.96e4·3-s + 2.62e5·4-s − 3.79e6·5-s − 1.00e7·6-s − 6.20e7·7-s + 1.34e8·8-s + 3.87e8·9-s − 1.94e9·10-s − 4.21e9·11-s − 5.15e9·12-s − 3.17e10·14-s + 7.46e10·15-s + 6.87e10·16-s + 1.98e11·18-s − 9.94e11·20-s + 1.22e12·21-s − 2.15e12·22-s − 2.64e12·24-s + 1.05e13·25-s − 7.62e12·27-s − 1.62e13·28-s − 2.05e12·29-s + 3.82e13·30-s − 1.79e13·31-s + 3.51e13·32-s + 8.29e13·33-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s + 4-s − 1.94·5-s − 6-s − 1.53·7-s + 8-s + 9-s − 1.94·10-s − 1.78·11-s − 12-s − 1.53·14-s + 1.94·15-s + 16-s + 18-s − 1.94·20-s + 1.53·21-s − 1.78·22-s − 24-s + 2.77·25-s − 27-s − 1.53·28-s − 0.141·29-s + 1.94·30-s − 0.677·31-s + 32-s + 1.78·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{19}{2})\) |
\(\approx\) |
\(0.6930064396\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6930064396\) |
| \(L(10)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{9} T \) |
| 3 | \( 1 + p^{9} T \) |
| good | 5 | \( 1 + 3792962 T + p^{18} T^{2} \) |
| 7 | \( 1 + 62062090 T + p^{18} T^{2} \) |
| 11 | \( 1 + 4213830710 T + p^{18} T^{2} \) |
| 13 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 17 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 19 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 23 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 29 | \( 1 + 2050619582450 T + p^{18} T^{2} \) |
| 31 | \( 1 + 17914374449242 T + p^{18} T^{2} \) |
| 37 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 41 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 43 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 47 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 53 | \( 1 + 2499566246756066 T + p^{18} T^{2} \) |
| 59 | \( 1 - 11981727959858890 T + p^{18} T^{2} \) |
| 61 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 67 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 71 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 73 | \( 1 + 502555043913550 T + p^{18} T^{2} \) |
| 79 | \( 1 - 57281163325307462 T + p^{18} T^{2} \) |
| 83 | \( 1 + 308366268689415494 T + p^{18} T^{2} \) |
| 89 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 97 | \( 1 - 390854313434426690 T + p^{18} T^{2} \) |
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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
−13.01608820503301182550217434800, −12.49498795901700714197823458711, −11.33863516591238008563025706762, −10.35661677242432929114261383625, −7.73868106005406849914602574082, −6.79529080069011390477701581765, −5.32391321907679460956075701142, −4.05026216673184795819849699308, −2.98490849016731633205391026660, −0.39977569999722499046106151864,
0.39977569999722499046106151864, 2.98490849016731633205391026660, 4.05026216673184795819849699308, 5.32391321907679460956075701142, 6.79529080069011390477701581765, 7.73868106005406849914602574082, 10.35661677242432929114261383625, 11.33863516591238008563025706762, 12.49498795901700714197823458711, 13.01608820503301182550217434800
