| L(s) = 1 | − 3·2-s + 3·3-s + 5·4-s + 6·5-s − 9·6-s − 2·7-s − 6·8-s + 7·9-s − 18·10-s + 3·11-s + 15·12-s − 10·13-s + 6·14-s + 18·15-s + 4·16-s − 6·17-s − 21·18-s − 3·19-s + 30·20-s − 6·21-s − 9·22-s − 18·24-s + 5·25-s + 30·26-s + 18·27-s − 10·28-s − 3·29-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 1.73·3-s + 5/2·4-s + 2.68·5-s − 3.67·6-s − 0.755·7-s − 2.12·8-s + 7/3·9-s − 5.69·10-s + 0.904·11-s + 4.33·12-s − 2.77·13-s + 1.60·14-s + 4.64·15-s + 16-s − 1.45·17-s − 4.94·18-s − 0.688·19-s + 6.70·20-s − 1.30·21-s − 1.91·22-s − 3.67·24-s + 25-s + 5.88·26-s + 3.46·27-s − 1.88·28-s − 0.557·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68574961 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68574961 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7359893723\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7359893723\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| good | 2 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} ) \) |
| 3 | $D_4\times C_2$ | \( 1 - p T + 2 T^{2} - p T^{3} + 13 T^{4} - p^{2} T^{5} + 2 p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) |
| 5 | $D_{4}$ | \( ( 1 - 3 T + 11 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 3 T - 4 T^{2} + 27 T^{3} - 51 T^{4} + 27 p T^{5} - 4 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 6 T + 13 T^{2} - 66 T^{3} - 372 T^{4} - 66 p T^{5} + 13 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 3 T - 20 T^{2} - 27 T^{3} + 309 T^{4} - 27 p T^{5} - 20 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 - 26 T^{2} + 147 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 3 T - 20 T^{2} - 3 p T^{3} - 9 p T^{4} - 3 p^{2} T^{5} - 20 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 6 T - 50 T^{2} + 24 T^{3} + 4239 T^{4} + 24 p T^{5} - 50 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 5 T + 34 T^{2} - 475 T^{3} - 2843 T^{4} - 475 p T^{5} + 34 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 89 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 - 12 T + 137 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 113 T^{2} + 9288 T^{4} - 113 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 12 T + 19 T^{2} + 108 T^{3} + 1488 T^{4} + 108 p T^{5} + 19 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_4\times C_2$ | \( 1 - 6 T + 10 T^{2} + 696 T^{3} - 6921 T^{4} + 696 p T^{5} + 10 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 121 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 21 T + 184 T^{2} + 1659 T^{3} + 18879 T^{4} + 1659 p T^{5} + 184 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 31 T + 538 T^{2} + 7099 T^{3} + 76303 T^{4} + 7099 p T^{5} + 538 p^{2} T^{6} + 31 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.27514006967594240862702061569, −9.962769946607979302328272250088, −9.659386416556933473818658366444, −9.607390863362311771317061218147, −9.348664020825086494977289568415, −9.265014826457146716128406306395, −8.709274848606384387222421860697, −8.492197260027017056801967848497, −8.402632954584516310066139809028, −8.006658353264053116592906854008, −7.32204083940866826485613760626, −7.20714144755062709813776420676, −6.90104654289965789026058436281, −6.70797965088300389274497248723, −6.42749491066853244813148334199, −5.95985015030157177617146234250, −5.36404778182236042436944321575, −5.21395122041646884883505643151, −4.41789987200615332892665062700, −4.18520982427647242351655117503, −3.47108356443198913441640188356, −2.61673524597116849145203397389, −2.42080627102110695534675623945, −2.02725007638130017726342416419, −1.80214490826245528977902820077,
1.80214490826245528977902820077, 2.02725007638130017726342416419, 2.42080627102110695534675623945, 2.61673524597116849145203397389, 3.47108356443198913441640188356, 4.18520982427647242351655117503, 4.41789987200615332892665062700, 5.21395122041646884883505643151, 5.36404778182236042436944321575, 5.95985015030157177617146234250, 6.42749491066853244813148334199, 6.70797965088300389274497248723, 6.90104654289965789026058436281, 7.20714144755062709813776420676, 7.32204083940866826485613760626, 8.006658353264053116592906854008, 8.402632954584516310066139809028, 8.492197260027017056801967848497, 8.709274848606384387222421860697, 9.265014826457146716128406306395, 9.348664020825086494977289568415, 9.607390863362311771317061218147, 9.659386416556933473818658366444, 9.962769946607979302328272250088, 10.27514006967594240862702061569
