| L(s) = 1 | + 6·3-s − 2·5-s + 21·9-s − 10·11-s − 2·13-s − 12·15-s − 3·17-s − 6·19-s + 23-s + 5·25-s + 54·27-s + 29-s + 8·31-s − 60·33-s − 3·37-s − 12·39-s − 3·41-s − 43-s − 42·45-s − 4·47-s − 18·51-s + 6·53-s + 20·55-s − 36·57-s − 5·59-s − 10·61-s + 4·65-s + ⋯ |
| L(s) = 1 | + 3.46·3-s − 0.894·5-s + 7·9-s − 3.01·11-s − 0.554·13-s − 3.09·15-s − 0.727·17-s − 1.37·19-s + 0.208·23-s + 25-s + 10.3·27-s + 0.185·29-s + 1.43·31-s − 10.4·33-s − 0.493·37-s − 1.92·39-s − 0.468·41-s − 0.152·43-s − 6.26·45-s − 0.583·47-s − 2.52·51-s + 0.824·53-s + 2.69·55-s − 4.76·57-s − 0.650·59-s − 1.28·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6492304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6492304 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.953504568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.953504568\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 4 T - 31 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 5 T - 34 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 11 T + 50 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 9 T - 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - T - 96 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−8.923781032136109116442443337297, −8.546453361308465543567623165293, −8.390234108052513344028370159535, −7.936638125327999264854198614958, −7.67852793221974912083613531525, −7.67701831751439767960377740049, −7.15112879071930192539196385682, −6.56897137624714705523716885909, −6.34209079238866235784715179709, −5.20981897835808206586528465806, −4.93748600024629697787379997784, −4.63590795681710940761514171071, −4.12969618109156314590874138866, −3.66027137218395960404644135829, −3.24271396632944491419866574990, −2.76188331593670224689884123995, −2.62352640094111975930607566512, −2.17413327539503297473762276189, −1.80764862383456807106165038138, −0.53352339913839038338613375499,
0.53352339913839038338613375499, 1.80764862383456807106165038138, 2.17413327539503297473762276189, 2.62352640094111975930607566512, 2.76188331593670224689884123995, 3.24271396632944491419866574990, 3.66027137218395960404644135829, 4.12969618109156314590874138866, 4.63590795681710940761514171071, 4.93748600024629697787379997784, 5.20981897835808206586528465806, 6.34209079238866235784715179709, 6.56897137624714705523716885909, 7.15112879071930192539196385682, 7.67701831751439767960377740049, 7.67852793221974912083613531525, 7.936638125327999264854198614958, 8.390234108052513344028370159535, 8.546453361308465543567623165293, 8.923781032136109116442443337297
