| L(s) = 1 | + 3·2-s + 3·3-s + 4·4-s − 2·5-s + 9·6-s − 4·7-s + 3·8-s + 2·9-s − 6·10-s − 7·11-s + 12·12-s + 10·13-s − 12·14-s − 6·15-s + 3·16-s − 12·17-s + 6·18-s − 8·20-s − 12·21-s − 21·22-s − 23-s + 9·24-s + 3·25-s + 30·26-s − 6·27-s − 16·28-s + 3·29-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 1.73·3-s + 2·4-s − 0.894·5-s + 3.67·6-s − 1.51·7-s + 1.06·8-s + 2/3·9-s − 1.89·10-s − 2.11·11-s + 3.46·12-s + 2.77·13-s − 3.20·14-s − 1.54·15-s + 3/4·16-s − 2.91·17-s + 1.41·18-s − 1.78·20-s − 2.61·21-s − 4.47·22-s − 0.208·23-s + 1.83·24-s + 3/5·25-s + 5.88·26-s − 1.15·27-s − 3.02·28-s + 0.557·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3258025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3258025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.864244438\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.864244438\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - p T + 7 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 7 T + 3 p T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + T + 45 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T + 49 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 15 T + 107 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 3 T + 65 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 43 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - T + 75 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 16 T + 153 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 9 T + 125 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 11 T + 117 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 93 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 173 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 146 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 133 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - T + 163 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−9.122541929476331572624286948149, −9.047443828937567089998162044729, −8.546786914046459803681617297868, −8.205402875684973851902928915518, −8.017593115350204332822419870162, −7.71588748756469928128942260479, −6.75852679799842559231476869128, −6.49231130161351995652205324752, −6.27352221603009970136942859038, −5.96046546573772756684310601278, −5.12835402075422783857760334767, −4.87588286437027067431880272499, −4.19476789550799328494919023721, −4.15754029474866119044372825238, −3.41215051482841317071662253214, −3.32766844767614596954852758652, −2.74243261929158210873429639762, −2.70341147120747461778621637303, −1.90447371278075048234866366222, −0.52979928575737552900917647218,
0.52979928575737552900917647218, 1.90447371278075048234866366222, 2.70341147120747461778621637303, 2.74243261929158210873429639762, 3.32766844767614596954852758652, 3.41215051482841317071662253214, 4.15754029474866119044372825238, 4.19476789550799328494919023721, 4.87588286437027067431880272499, 5.12835402075422783857760334767, 5.96046546573772756684310601278, 6.27352221603009970136942859038, 6.49231130161351995652205324752, 6.75852679799842559231476869128, 7.71588748756469928128942260479, 8.017593115350204332822419870162, 8.205402875684973851902928915518, 8.546786914046459803681617297868, 9.047443828937567089998162044729, 9.122541929476331572624286948149
