| L(s) = 1 | + 3·2-s + 3·3-s + 4·4-s + 2·5-s + 9·6-s + 4·7-s + 3·8-s + 2·9-s + 6·10-s − 5·11-s + 12·12-s + 5·13-s + 12·14-s + 6·15-s + 3·16-s + 17-s + 6·18-s − 2·19-s + 8·20-s + 12·21-s − 15·22-s + 11·23-s + 9·24-s − 2·25-s + 15·26-s − 6·27-s + 16·28-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 1.73·3-s + 2·4-s + 0.894·5-s + 3.67·6-s + 1.51·7-s + 1.06·8-s + 2/3·9-s + 1.89·10-s − 1.50·11-s + 3.46·12-s + 1.38·13-s + 3.20·14-s + 1.54·15-s + 3/4·16-s + 0.242·17-s + 1.41·18-s − 0.458·19-s + 1.78·20-s + 2.61·21-s − 3.19·22-s + 2.29·23-s + 1.83·24-s − 2/5·25-s + 2.94·26-s − 1.15·27-s + 3.02·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3418801 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3418801 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(20.50872522\) |
| \(L(\frac12)\) |
\(\approx\) |
\(20.50872522\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 43 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - p T + 7 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 4 T + 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 5 T + 27 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 5 T + 31 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 11 T + 75 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 3 T + 65 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 10 T + 87 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 9 T + 103 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 5 T + 111 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + T + 87 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - T + 111 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + T + 123 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 3 T - 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 3 T + 137 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 5 T + 163 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 3 T - 43 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 3 T + 169 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 238 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.157245378146195015630120050202, −9.080097323754220318811934853276, −8.541359632058773430479711239536, −8.287195606222105755667744999983, −7.80978719355560733355832269465, −7.57359826267334166325604059190, −7.21980409670916809098736502934, −6.37460183360299329862601038479, −5.93813133842221850413759612647, −5.68297789683499467164254856664, −5.25615959319269638699250998013, −5.01667413736422700598329710219, −4.41697556712553738165567929749, −4.14390511826446518214148393203, −3.43311702555913566159419771925, −3.28920948309274159165405594782, −2.55895183636128467780974768368, −2.48114181530850170984026483376, −1.77093037278822331842001548965, −1.15347601570830875651791221785,
1.15347601570830875651791221785, 1.77093037278822331842001548965, 2.48114181530850170984026483376, 2.55895183636128467780974768368, 3.28920948309274159165405594782, 3.43311702555913566159419771925, 4.14390511826446518214148393203, 4.41697556712553738165567929749, 5.01667413736422700598329710219, 5.25615959319269638699250998013, 5.68297789683499467164254856664, 5.93813133842221850413759612647, 6.37460183360299329862601038479, 7.21980409670916809098736502934, 7.57359826267334166325604059190, 7.80978719355560733355832269465, 8.287195606222105755667744999983, 8.541359632058773430479711239536, 9.080097323754220318811934853276, 9.157245378146195015630120050202
