| L(s) = 1 | − 4·2-s + 8·4-s + 2·5-s − 8·8-s − 8·10-s − 6·11-s − 4·16-s + 16·20-s + 24·22-s + 3·25-s + 32·32-s − 16·40-s − 22·41-s + 8·43-s − 48·44-s + 8·47-s − 49-s − 12·50-s − 12·55-s − 24·59-s + 26·61-s − 64·64-s + 10·71-s + 26·79-s − 8·80-s + 88·82-s − 12·83-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 4·4-s + 0.894·5-s − 2.82·8-s − 2.52·10-s − 1.80·11-s − 16-s + 3.57·20-s + 5.11·22-s + 3/5·25-s + 5.65·32-s − 2.52·40-s − 3.43·41-s + 1.21·43-s − 7.23·44-s + 1.16·47-s − 1/7·49-s − 1.69·50-s − 1.61·55-s − 3.12·59-s + 3.32·61-s − 8·64-s + 1.18·71-s + 2.92·79-s − 0.894·80-s + 9.71·82-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57836025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57836025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 21 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 33 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 61 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 181 T^{2} + p^{2} T^{4} \) |
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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
−7.81124073047152845656480205099, −7.68040116864916347266982244689, −7.05569899572073300451950979648, −7.02168106314143066184978822425, −6.42000379442205464605399726871, −6.38846259779536639259979972351, −5.55208993339332068926075537175, −5.40136591153822374794241899561, −4.92232800471427199320366956944, −4.76204396221889242899003029145, −4.08821157415317262881803808505, −3.56871020078579112707395372805, −2.99285670264988643663354062969, −2.51101442625650056711311197751, −2.08902286492655227701855372944, −2.03154918035443994284212611814, −1.19375967252927720352670646678, −0.995700645392042598707326917835, 0, 0,
0.995700645392042598707326917835, 1.19375967252927720352670646678, 2.03154918035443994284212611814, 2.08902286492655227701855372944, 2.51101442625650056711311197751, 2.99285670264988643663354062969, 3.56871020078579112707395372805, 4.08821157415317262881803808505, 4.76204396221889242899003029145, 4.92232800471427199320366956944, 5.40136591153822374794241899561, 5.55208993339332068926075537175, 6.38846259779536639259979972351, 6.42000379442205464605399726871, 7.02168106314143066184978822425, 7.05569899572073300451950979648, 7.68040116864916347266982244689, 7.81124073047152845656480205099
