| L(s) = 1 | − 3-s + 5-s + 2·7-s − 2·9-s − 6·13-s − 15-s + 2·17-s − 6·19-s − 2·21-s − 6·25-s + 2·27-s + 5·31-s + 2·35-s − 2·37-s + 6·39-s − 3·41-s − 11·43-s − 2·45-s + 2·47-s + 3·49-s − 2·51-s + 11·53-s + 6·57-s − 4·59-s + 5·61-s − 4·63-s − 6·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s − 2/3·9-s − 1.66·13-s − 0.258·15-s + 0.485·17-s − 1.37·19-s − 0.436·21-s − 6/5·25-s + 0.384·27-s + 0.898·31-s + 0.338·35-s − 0.328·37-s + 0.960·39-s − 0.468·41-s − 1.67·43-s − 0.298·45-s + 0.291·47-s + 3/7·49-s − 0.280·51-s + 1.51·53-s + 0.794·57-s − 0.520·59-s + 0.640·61-s − 0.503·63-s − 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58003456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58003456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - T + 7 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 5 T + 65 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 62 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 55 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 11 T + 113 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 82 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 11 T + 133 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 5 T + 99 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 7 T + 143 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 5 T + 123 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 18 T + 226 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 26 T + 334 T^{2} + 26 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 9 T + 211 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
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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
−7.63016014022101103487022071276, −7.38988856810408992250970188395, −6.88504614651688014263683805649, −6.77770314294095575671931760564, −6.10401489682865978310945972154, −6.09004899815329670739141928375, −5.44876031132231184750358307196, −5.34568633083706373941449836479, −4.92131998321942959458128814767, −4.71567236984094831812065547465, −4.03485968359762018094956987200, −3.95725226192777045142416743743, −3.32315965565525780387708938368, −2.67231087050597624240718026027, −2.48137675798750353652743958826, −2.06699032090843410407735925647, −1.59414313682064723442830397276, −1.00196639664323307928118727823, 0, 0,
1.00196639664323307928118727823, 1.59414313682064723442830397276, 2.06699032090843410407735925647, 2.48137675798750353652743958826, 2.67231087050597624240718026027, 3.32315965565525780387708938368, 3.95725226192777045142416743743, 4.03485968359762018094956987200, 4.71567236984094831812065547465, 4.92131998321942959458128814767, 5.34568633083706373941449836479, 5.44876031132231184750358307196, 6.09004899815329670739141928375, 6.10401489682865978310945972154, 6.77770314294095575671931760564, 6.88504614651688014263683805649, 7.38988856810408992250970188395, 7.63016014022101103487022071276
