| L(s) = 1 | + 2-s + 4-s + 2·8-s − 2·11-s + 2·16-s − 2·22-s + 4·23-s + 2·25-s − 2·29-s + 2·32-s + 37-s + 43-s − 2·44-s + 4·46-s + 2·50-s + 53-s − 2·58-s + 3·64-s + 67-s − 2·71-s + 74-s + 79-s + 86-s − 4·88-s + 4·92-s + 2·100-s + 106-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s + 2·8-s − 2·11-s + 2·16-s − 2·22-s + 4·23-s + 2·25-s − 2·29-s + 2·32-s + 37-s + 43-s − 2·44-s + 4·46-s + 2·50-s + 53-s − 2·58-s + 3·64-s + 67-s − 2·71-s + 74-s + 79-s + 86-s − 4·88-s + 4·92-s + 2·100-s + 106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15752961 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15752961 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.617407375\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.617407375\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 29 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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
−8.697473754046915021148215257094, −8.499304604711944762117605735445, −7.73105202732569175094461163449, −7.67288246382947113326919806255, −7.20293210164341366394866019728, −7.19890554288722166973373907464, −6.64082066924560171453486413432, −6.26603183857101079673955835848, −5.57906746781731957707024349417, −5.23499533780897345844950222898, −5.20777323485591109465663931393, −4.71866626248750901945578363804, −4.51188957817516427714565998638, −3.79251538040541929956584639675, −3.41318230797011177309638363741, −2.89587841624601740402207729696, −2.46953940453351008360053987677, −2.39922604642672127674618628308, −1.21359316281921936916569736831, −1.12264797767380351547029317859,
1.12264797767380351547029317859, 1.21359316281921936916569736831, 2.39922604642672127674618628308, 2.46953940453351008360053987677, 2.89587841624601740402207729696, 3.41318230797011177309638363741, 3.79251538040541929956584639675, 4.51188957817516427714565998638, 4.71866626248750901945578363804, 5.20777323485591109465663931393, 5.23499533780897345844950222898, 5.57906746781731957707024349417, 6.26603183857101079673955835848, 6.64082066924560171453486413432, 7.19890554288722166973373907464, 7.20293210164341366394866019728, 7.67288246382947113326919806255, 7.73105202732569175094461163449, 8.499304604711944762117605735445, 8.697473754046915021148215257094
