| L(s) = 1 | − 2-s + 3·3-s + 2·5-s − 3·6-s + 8-s + 3·9-s − 2·10-s + 11-s + 14·13-s + 6·15-s − 16-s + 2·17-s − 3·18-s − 22-s + 8·23-s + 3·24-s + 5·25-s − 14·26-s − 10·29-s − 6·30-s + 4·31-s + 3·33-s − 2·34-s − 4·37-s + 42·39-s + 2·40-s − 8·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s + 0.894·5-s − 1.22·6-s + 0.353·8-s + 9-s − 0.632·10-s + 0.301·11-s + 3.88·13-s + 1.54·15-s − 1/4·16-s + 0.485·17-s − 0.707·18-s − 0.213·22-s + 1.66·23-s + 0.612·24-s + 25-s − 2.74·26-s − 1.85·29-s − 1.09·30-s + 0.718·31-s + 0.522·33-s − 0.342·34-s − 0.657·37-s + 6.72·39-s + 0.316·40-s − 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.219728854\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.219728854\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| 11 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 9 T + 14 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 4 T - 57 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 9 T + 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
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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
−9.987249233724067673064911618718, −9.417711663989457671188794358601, −9.012392425318081695402550735781, −8.754960278265872074631688722312, −8.728933127791746656051029044292, −8.174738755820159937297751641048, −8.053713790933563715762567808887, −7.20943612756684122535948040317, −6.75857079352251742256897417307, −6.52071234769339102716149612620, −5.71995438251888922625291566644, −5.65439655256188949537613259259, −4.91470121489967575000501622686, −4.09966570848640503506198136206, −3.64921371531177322911806965613, −3.16524021314363172304828328656, −3.08261515511990959137095661411, −1.97231368035948608078014724565, −1.49622510661743195729973416429, −1.10675352295600983248161559889,
1.10675352295600983248161559889, 1.49622510661743195729973416429, 1.97231368035948608078014724565, 3.08261515511990959137095661411, 3.16524021314363172304828328656, 3.64921371531177322911806965613, 4.09966570848640503506198136206, 4.91470121489967575000501622686, 5.65439655256188949537613259259, 5.71995438251888922625291566644, 6.52071234769339102716149612620, 6.75857079352251742256897417307, 7.20943612756684122535948040317, 8.053713790933563715762567808887, 8.174738755820159937297751641048, 8.728933127791746656051029044292, 8.754960278265872074631688722312, 9.012392425318081695402550735781, 9.417711663989457671188794358601, 9.987249233724067673064911618718
