| L(s) = 1 | + 2·2-s + 3·4-s − 4·5-s − 2·7-s + 4·8-s − 8·10-s − 4·13-s − 4·14-s + 5·16-s + 4·17-s − 12·20-s + 2·23-s + 4·25-s − 8·26-s − 6·28-s + 4·29-s − 4·31-s + 6·32-s + 8·34-s + 8·35-s − 12·37-s − 16·40-s − 4·41-s − 16·43-s + 4·46-s − 12·47-s + 3·49-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 1.78·5-s − 0.755·7-s + 1.41·8-s − 2.52·10-s − 1.10·13-s − 1.06·14-s + 5/4·16-s + 0.970·17-s − 2.68·20-s + 0.417·23-s + 4/5·25-s − 1.56·26-s − 1.13·28-s + 0.742·29-s − 0.718·31-s + 1.06·32-s + 1.37·34-s + 1.35·35-s − 1.97·37-s − 2.52·40-s − 0.624·41-s − 2.43·43-s + 0.589·46-s − 1.75·47-s + 3/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8398404 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8398404 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 92 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 16 T + 148 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 124 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 20 T + 204 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 16 T + 180 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 20 T + 234 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 16 T + 228 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 222 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−8.497668025166204841908600239424, −7.961582039728902370992880216159, −7.66793513184510774437727074531, −7.48269768193583810292924626430, −6.95918620159432844378019038757, −6.71586396376731562773253214748, −6.34174944582184202673289721156, −5.86215601795926089456645120576, −5.21243966038059300988981665105, −5.07565309960027264421075525419, −4.67598034148736936435469450256, −4.20066138958227165166533664953, −3.81124544416974092611957787235, −3.28788660917471099105422629756, −3.02306364628432693855753026805, −2.96691563900250279337824114837, −1.71505190828445917929709778092, −1.57605959095326722176577783300, 0, 0,
1.57605959095326722176577783300, 1.71505190828445917929709778092, 2.96691563900250279337824114837, 3.02306364628432693855753026805, 3.28788660917471099105422629756, 3.81124544416974092611957787235, 4.20066138958227165166533664953, 4.67598034148736936435469450256, 5.07565309960027264421075525419, 5.21243966038059300988981665105, 5.86215601795926089456645120576, 6.34174944582184202673289721156, 6.71586396376731562773253214748, 6.95918620159432844378019038757, 7.48269768193583810292924626430, 7.66793513184510774437727074531, 7.961582039728902370992880216159, 8.497668025166204841908600239424
