| L(s) = 1 | − 4-s − 9-s + 16-s + 16·19-s + 4·29-s − 16·31-s + 36-s + 20·41-s + 14·49-s + 8·59-s + 4·61-s − 64-s − 16·76-s − 16·79-s + 81-s − 36·89-s − 36·101-s − 4·109-s − 4·116-s − 22·121-s + 16·124-s + 127-s + 131-s + 137-s + 139-s − 144-s + 149-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s + 1/4·16-s + 3.67·19-s + 0.742·29-s − 2.87·31-s + 1/6·36-s + 3.12·41-s + 2·49-s + 1.04·59-s + 0.512·61-s − 1/8·64-s − 1.83·76-s − 1.80·79-s + 1/9·81-s − 3.81·89-s − 3.58·101-s − 0.383·109-s − 0.371·116-s − 2·121-s + 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.0833·144-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11902500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.320394019\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.320394019\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 23 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 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
−8.978778197193570420699576292967, −8.263040115975477844141059648539, −8.128037123783661624668143412407, −7.54361292084043466632852199412, −7.37894055051468767648698997355, −7.00343132687240832047665025288, −6.78066416556535522759105040992, −5.78758948220945861165345498537, −5.64904726556033708389340789617, −5.39734590847341460205009941519, −5.34850392393690480443788472287, −4.30454558672118918730803722855, −4.27958047405848625276311608681, −3.70439557043405014615860680480, −3.28336949575985856132853825805, −2.65159171267785984272085719989, −2.62787488954381454945334269874, −1.51521970497318389407268563246, −1.16683217121102224880516085635, −0.52844274799798813899962652835,
0.52844274799798813899962652835, 1.16683217121102224880516085635, 1.51521970497318389407268563246, 2.62787488954381454945334269874, 2.65159171267785984272085719989, 3.28336949575985856132853825805, 3.70439557043405014615860680480, 4.27958047405848625276311608681, 4.30454558672118918730803722855, 5.34850392393690480443788472287, 5.39734590847341460205009941519, 5.64904726556033708389340789617, 5.78758948220945861165345498537, 6.78066416556535522759105040992, 7.00343132687240832047665025288, 7.37894055051468767648698997355, 7.54361292084043466632852199412, 8.128037123783661624668143412407, 8.263040115975477844141059648539, 8.978778197193570420699576292967
