| L(s) = 1 | + 2·2-s + 4-s + 3·5-s + 5·7-s − 2·9-s + 6·10-s − 13-s + 10·14-s + 16-s + 3·17-s − 4·18-s − 6·19-s + 3·20-s − 23-s + 2·25-s − 2·26-s + 5·28-s + 4·29-s − 31-s − 2·32-s + 6·34-s + 15·35-s − 2·36-s − 12·38-s + 11·41-s + 2·43-s − 6·45-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s + 1.34·5-s + 1.88·7-s − 2/3·9-s + 1.89·10-s − 0.277·13-s + 2.67·14-s + 1/4·16-s + 0.727·17-s − 0.942·18-s − 1.37·19-s + 0.670·20-s − 0.208·23-s + 2/5·25-s − 0.392·26-s + 0.944·28-s + 0.742·29-s − 0.179·31-s − 0.353·32-s + 1.02·34-s + 2.53·35-s − 1/3·36-s − 1.94·38-s + 1.71·41-s + 0.304·43-s − 0.894·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143065 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143065 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.558120839\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.558120839\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 71 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3 T + 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 11 T + 60 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T - 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 3 T + 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + T - 40 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + T + 30 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + T + 28 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 + 7 T + 40 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 15 T + 108 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
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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
−13.7725586493, −13.4897211771, −12.8517686457, −12.6943498452, −12.1202107775, −11.6861282205, −11.0841623252, −10.8971925885, −10.3647960913, −9.82421104523, −9.36266948396, −8.70071370189, −8.40229180084, −7.78835505944, −7.45841539977, −6.52607036785, −6.05322750580, −5.63235634267, −5.21686946097, −4.66710234380, −4.41684217270, −3.66291664492, −2.72800253696, −2.09632441663, −1.44162997443,
1.44162997443, 2.09632441663, 2.72800253696, 3.66291664492, 4.41684217270, 4.66710234380, 5.21686946097, 5.63235634267, 6.05322750580, 6.52607036785, 7.45841539977, 7.78835505944, 8.40229180084, 8.70071370189, 9.36266948396, 9.82421104523, 10.3647960913, 10.8971925885, 11.0841623252, 11.6861282205, 12.1202107775, 12.6943498452, 12.8517686457, 13.4897211771, 13.7725586493
