| L(s) = 1 | + 4·2-s + 4-s + 26·5-s − 10·7-s − 24·8-s + 104·10-s + 60·11-s − 24·13-s − 40·14-s − 47·16-s + 150·17-s − 46·19-s + 26·20-s + 240·22-s + 46·23-s + 262·25-s − 96·26-s − 10·28-s + 216·29-s + 324·31-s − 52·32-s + 600·34-s − 260·35-s + 140·37-s − 184·38-s − 624·40-s + 364·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/8·4-s + 2.32·5-s − 0.539·7-s − 1.06·8-s + 3.28·10-s + 1.64·11-s − 0.512·13-s − 0.763·14-s − 0.734·16-s + 2.14·17-s − 0.555·19-s + 0.290·20-s + 2.32·22-s + 0.417·23-s + 2.09·25-s − 0.724·26-s − 0.0674·28-s + 1.38·29-s + 1.87·31-s − 0.287·32-s + 3.02·34-s − 1.25·35-s + 0.622·37-s − 0.785·38-s − 2.46·40-s + 1.38·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42849 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42849 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.412509437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.412509437\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - p^{2} T + 15 T^{2} - p^{5} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 26 T + 414 T^{2} - 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 10 T + 466 T^{2} + 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 60 T + 3062 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 24 T + 38 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 150 T + 15446 T^{2} - 150 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 46 T + 8802 T^{2} + 46 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 216 T + 45862 T^{2} - 216 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 324 T + 55406 T^{2} - 324 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 140 T + 85726 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 364 T + 150486 T^{2} - 364 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 126 T + 107858 T^{2} + 126 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 120 T - 96274 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 490 T + 357174 T^{2} + 490 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 32 T + 141894 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 908 T + 512158 T^{2} + 908 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 874 T + 792370 T^{2} + 874 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 488 T + 627438 T^{2} + 488 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 652 T + 525190 T^{2} - 652 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1198 T + 1212034 T^{2} + 1198 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 100 T + 1112454 T^{2} - 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 102 T + 724334 T^{2} + 102 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1624 T + 1858110 T^{2} - 1624 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−12.26473734384424517465870120536, −12.08855167355999269154710049814, −11.49699498622575203397551005523, −10.53664978995765250686484920247, −9.978863069529243581178880349723, −9.907674480678983152732411617577, −9.219629653375036728487758313520, −9.147321816479382648611307364740, −8.227249603961742161210678483528, −7.52161288799196957430089060441, −6.40677883827513762853864538916, −6.40052265229256585779294249322, −5.88113510193971074688276552029, −5.42315269279550347711199280603, −4.58646275519706348734053763090, −4.40313437752440134788302497308, −3.18733937681174670635732516026, −2.97724174210892942791621363881, −1.76792535509813731041122588356, −1.04536357832079579036922131738,
1.04536357832079579036922131738, 1.76792535509813731041122588356, 2.97724174210892942791621363881, 3.18733937681174670635732516026, 4.40313437752440134788302497308, 4.58646275519706348734053763090, 5.42315269279550347711199280603, 5.88113510193971074688276552029, 6.40052265229256585779294249322, 6.40677883827513762853864538916, 7.52161288799196957430089060441, 8.227249603961742161210678483528, 9.147321816479382648611307364740, 9.219629653375036728487758313520, 9.907674480678983152732411617577, 9.978863069529243581178880349723, 10.53664978995765250686484920247, 11.49699498622575203397551005523, 12.08855167355999269154710049814, 12.26473734384424517465870120536
