| L(s) = 1 | + 2·2-s − 3-s + 3·4-s − 5-s − 2·6-s + 2·7-s + 4·8-s − 2·10-s − 3·12-s + 4·14-s + 15-s + 5·16-s − 3·20-s − 2·21-s + 23-s − 4·24-s + 27-s + 6·28-s − 2·29-s + 2·30-s + 6·32-s − 2·35-s − 4·40-s − 2·41-s − 4·42-s + 43-s + 2·46-s + ⋯ |
| L(s) = 1 | + 2·2-s − 3-s + 3·4-s − 5-s − 2·6-s + 2·7-s + 4·8-s − 2·10-s − 3·12-s + 4·14-s + 15-s + 5·16-s − 3·20-s − 2·21-s + 23-s − 4·24-s + 27-s + 6·28-s − 2·29-s + 2·30-s + 6·32-s − 2·35-s − 4·40-s − 2·41-s − 4·42-s + 43-s + 2·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.978677548\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.978677548\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−10.56333807576407540861189283847, −10.03946438060623581981675063067, −9.199345189460336447250142584786, −8.734327343659711394405046560764, −8.144522968151241496210575590254, −7.83574090911743899413784034466, −7.51512761469919996600133553583, −7.16706829018412329478891761561, −6.65299863980579408457836536642, −6.21028317959422507123164569761, −5.58393014331547726114575068556, −5.44166710902342945117136858834, −4.91267258189623442618300712247, −4.70356903545416360686776974928, −4.20712617639826792368601817554, −3.78438045340965115522055707355, −3.16649378421366817175115481493, −2.65889593714489926532240422109, −1.64881591307567112670085305000, −1.51842989733061180236391651852,
1.51842989733061180236391651852, 1.64881591307567112670085305000, 2.65889593714489926532240422109, 3.16649378421366817175115481493, 3.78438045340965115522055707355, 4.20712617639826792368601817554, 4.70356903545416360686776974928, 4.91267258189623442618300712247, 5.44166710902342945117136858834, 5.58393014331547726114575068556, 6.21028317959422507123164569761, 6.65299863980579408457836536642, 7.16706829018412329478891761561, 7.51512761469919996600133553583, 7.83574090911743899413784034466, 8.144522968151241496210575590254, 8.734327343659711394405046560764, 9.199345189460336447250142584786, 10.03946438060623581981675063067, 10.56333807576407540861189283847
