| L(s) = 1 | − 4·3-s − 4-s + 6·9-s + 4·12-s + 16-s − 12·17-s + 10·25-s + 4·27-s − 12·29-s − 6·36-s − 16·43-s − 4·48-s − 49-s + 48·51-s + 12·53-s + 16·61-s − 64-s + 12·68-s − 40·75-s + 16·79-s − 37·81-s + 48·87-s − 10·100-s + 8·103-s + 24·107-s − 4·108-s + 12·113-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 1/2·4-s + 2·9-s + 1.15·12-s + 1/4·16-s − 2.91·17-s + 2·25-s + 0.769·27-s − 2.22·29-s − 36-s − 2.43·43-s − 0.577·48-s − 1/7·49-s + 6.72·51-s + 1.64·53-s + 2.04·61-s − 1/8·64-s + 1.45·68-s − 4.61·75-s + 1.80·79-s − 4.11·81-s + 5.14·87-s − 100-s + 0.788·103-s + 2.32·107-s − 0.384·108-s + 1.12·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5597956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5597956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5405929312\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5405929312\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| 13 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−9.052175056436439656075028159780, −8.755073655206666995165644412870, −8.493651615787648405073551393837, −8.266052831580236057891589273785, −7.24216176980540627275311900577, −7.04422254691782314558920018593, −6.82059044630213788396878470942, −6.44260788207622594752369395737, −5.91776663504495619727237258278, −5.72743885041659002539839945793, −5.19955081039235167491220060446, −4.81839409357725726974604782082, −4.67669110009385221598947651279, −4.21629439143388052701881996850, −3.50990628967231016160298573793, −3.08975239285096808873310112432, −2.13910727694720374698679071715, −1.91399277806054882634561597237, −0.64897985903083337367480226676, −0.50416831819487133808294935056,
0.50416831819487133808294935056, 0.64897985903083337367480226676, 1.91399277806054882634561597237, 2.13910727694720374698679071715, 3.08975239285096808873310112432, 3.50990628967231016160298573793, 4.21629439143388052701881996850, 4.67669110009385221598947651279, 4.81839409357725726974604782082, 5.19955081039235167491220060446, 5.72743885041659002539839945793, 5.91776663504495619727237258278, 6.44260788207622594752369395737, 6.82059044630213788396878470942, 7.04422254691782314558920018593, 7.24216176980540627275311900577, 8.266052831580236057891589273785, 8.493651615787648405073551393837, 8.755073655206666995165644412870, 9.052175056436439656075028159780
