| L(s) = 1 | − 4·2-s + 12·4-s − 32·8-s + 80·16-s − 124·17-s + 168·19-s − 280·23-s + 32·31-s − 192·32-s + 496·34-s − 672·38-s + 1.12e3·46-s − 200·47-s − 190·49-s − 1.47e3·53-s − 716·61-s − 128·62-s + 448·64-s − 1.48e3·68-s + 2.01e3·76-s − 1.87e3·79-s − 2.60e3·83-s − 3.36e3·92-s + 800·94-s + 760·98-s + 5.90e3·106-s − 1.39e3·107-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s + 5/4·16-s − 1.76·17-s + 2.02·19-s − 2.53·23-s + 0.185·31-s − 1.06·32-s + 2.50·34-s − 2.86·38-s + 3.58·46-s − 0.620·47-s − 0.553·49-s − 3.82·53-s − 1.50·61-s − 0.262·62-s + 7/8·64-s − 2.65·68-s + 3.04·76-s − 2.66·79-s − 3.44·83-s − 3.80·92-s + 0.877·94-s + 0.783·98-s + 5.40·106-s − 1.25·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 190 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2166 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 70 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 62 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 84 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 140 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 8602 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 16 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 41290 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 88242 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 32038 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 100 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 738 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 6378 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 358 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 114698 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 159122 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 579634 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 936 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 1304 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 902034 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 1251970 T^{2} + p^{6} T^{4} \) |
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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
−10.18765707304720792818169879095, −9.995331697422670902643257728420, −9.379152043451585869929183772079, −9.378618340451525220829790059979, −8.582745033768608322351103149210, −8.232310784625939957759910238909, −7.73287156230660741353312843302, −7.50258883996243104474949706481, −6.73401513373918840503327020389, −6.44061297607408878022489818872, −5.90180250566508759535125418941, −5.35552157243366734246993260832, −4.53057780179287978974527030595, −4.06527427393672047564010459040, −3.08357769040920168222943060893, −2.75514571384361216448314486432, −1.66663760789989390713407388529, −1.50802647394342089360616120975, 0, 0,
1.50802647394342089360616120975, 1.66663760789989390713407388529, 2.75514571384361216448314486432, 3.08357769040920168222943060893, 4.06527427393672047564010459040, 4.53057780179287978974527030595, 5.35552157243366734246993260832, 5.90180250566508759535125418941, 6.44061297607408878022489818872, 6.73401513373918840503327020389, 7.50258883996243104474949706481, 7.73287156230660741353312843302, 8.232310784625939957759910238909, 8.582745033768608322351103149210, 9.378618340451525220829790059979, 9.379152043451585869929183772079, 9.995331697422670902643257728420, 10.18765707304720792818169879095
