| L(s) = 1 | + 2·2-s + 3·4-s − 7-s + 4·8-s + 9-s + 2·11-s − 2·14-s + 5·16-s + 2·18-s + 4·22-s − 6·25-s − 3·28-s − 4·29-s + 6·32-s + 3·36-s − 4·37-s + 8·43-s + 6·44-s + 49-s − 12·50-s − 4·53-s − 4·56-s − 8·58-s − 63-s + 7·64-s + 24·67-s + 16·71-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.377·7-s + 1.41·8-s + 1/3·9-s + 0.603·11-s − 0.534·14-s + 5/4·16-s + 0.471·18-s + 0.852·22-s − 6/5·25-s − 0.566·28-s − 0.742·29-s + 1.06·32-s + 1/2·36-s − 0.657·37-s + 1.21·43-s + 0.904·44-s + 1/7·49-s − 1.69·50-s − 0.549·53-s − 0.534·56-s − 1.05·58-s − 0.125·63-s + 7/8·64-s + 2.93·67-s + 1.89·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1494108 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1494108 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.621590498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.621590498\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( 1 + T \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−7.68971915466515831370427459157, −7.16036475066804637810786253712, −7.14524003659077370950096225115, −6.34557563328331303195978146208, −6.24570659632441259030662106209, −5.62678747313723194419784671277, −5.33211397644666962563070735994, −4.77334097811356740007680534766, −4.18774926881798203669654007783, −3.92094938594450734186502076294, −3.44789602839468310880410655927, −2.96523175330119569799799482797, −2.10377029048710954694042998887, −1.88425382150531611231140922836, −0.801691090903374504918552012961,
0.801691090903374504918552012961, 1.88425382150531611231140922836, 2.10377029048710954694042998887, 2.96523175330119569799799482797, 3.44789602839468310880410655927, 3.92094938594450734186502076294, 4.18774926881798203669654007783, 4.77334097811356740007680534766, 5.33211397644666962563070735994, 5.62678747313723194419784671277, 6.24570659632441259030662106209, 6.34557563328331303195978146208, 7.14524003659077370950096225115, 7.16036475066804637810786253712, 7.68971915466515831370427459157
