| L(s) = 1 | + 2-s − 4-s − 4·7-s − 3·8-s + 2·9-s − 2·11-s − 4·14-s − 16-s + 2·18-s − 2·22-s + 12·23-s + 2·25-s + 4·28-s − 14·29-s + 5·32-s − 2·36-s + 4·37-s − 8·43-s + 2·44-s + 12·46-s + 9·49-s + 2·50-s + 6·53-s + 12·56-s − 14·58-s − 8·63-s + 7·64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.51·7-s − 1.06·8-s + 2/3·9-s − 0.603·11-s − 1.06·14-s − 1/4·16-s + 0.471·18-s − 0.426·22-s + 2.50·23-s + 2/5·25-s + 0.755·28-s − 2.59·29-s + 0.883·32-s − 1/3·36-s + 0.657·37-s − 1.21·43-s + 0.301·44-s + 1.76·46-s + 9/7·49-s + 0.282·50-s + 0.824·53-s + 1.60·56-s − 1.83·58-s − 1.00·63-s + 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 41 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
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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
−7.56733619312816819531740305261, −7.20421653497416341972870084569, −6.85552111025605262181450181804, −6.51007540812857910327410929642, −5.75706051397583714929402063481, −5.60862750508571607983732563032, −5.12874623491498510803274444388, −4.55793246810234766003248193565, −4.15004975073289992247147290174, −3.53748492184856308530402080877, −3.14323039296420278130805486479, −2.84497367648824844588446125252, −1.96342256826295179045814750714, −0.933505167233924177873085894208, 0,
0.933505167233924177873085894208, 1.96342256826295179045814750714, 2.84497367648824844588446125252, 3.14323039296420278130805486479, 3.53748492184856308530402080877, 4.15004975073289992247147290174, 4.55793246810234766003248193565, 5.12874623491498510803274444388, 5.60862750508571607983732563032, 5.75706051397583714929402063481, 6.51007540812857910327410929642, 6.85552111025605262181450181804, 7.20421653497416341972870084569, 7.56733619312816819531740305261
