| L(s) = 1 | + 2·2-s + 4-s + 24·5-s + 12·8-s + 48·10-s − 44·11-s − 11·16-s − 164·17-s − 48·19-s + 24·20-s − 88·22-s − 8·23-s + 238·25-s − 404·29-s − 40·31-s − 170·32-s − 328·34-s + 100·37-s − 96·38-s + 288·40-s + 200·41-s − 616·43-s − 44·44-s − 16·46-s − 324·47-s − 630·49-s + 476·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/8·4-s + 2.14·5-s + 0.530·8-s + 1.51·10-s − 1.20·11-s − 0.171·16-s − 2.33·17-s − 0.579·19-s + 0.268·20-s − 0.852·22-s − 0.0725·23-s + 1.90·25-s − 2.58·29-s − 0.231·31-s − 0.939·32-s − 1.65·34-s + 0.444·37-s − 0.409·38-s + 1.13·40-s + 0.761·41-s − 2.18·43-s − 0.150·44-s − 0.0512·46-s − 1.00·47-s − 1.83·49-s + 1.34·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - p T + 3 T^{2} - p^{4} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 24 T + 338 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 90 p T^{2} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 p T + 1130 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 164 T + 16326 T^{2} + 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 48 T + 14238 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 8 T - 7906 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 404 T + 81518 T^{2} + 404 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 40 T + 50518 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 100 T + 92830 T^{2} - 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 200 T + 24138 T^{2} - 200 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 616 T + 216022 T^{2} + 616 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 324 T + 219554 T^{2} + 324 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 164 T + 102878 T^{2} - 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 140 T + 393258 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 628 T + 293614 T^{2} - 628 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 472 T + 252622 T^{2} - 472 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 428 T + 662834 T^{2} - 428 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 900 T + 899670 T^{2} - 900 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 432 T + 924318 T^{2} + 432 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1388 T + 1567866 T^{2} + 1388 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 960 T + 1134938 T^{2} - 960 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 532 T + 1218502 T^{2} - 532 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.195253064873447991330262613729, −8.458460268400165322467861212245, −8.168851354407513143474785003809, −7.73379965291232005433046568282, −6.94452044853434994534463211027, −6.78172647251795419502247490688, −6.51428847088952488083938286690, −5.88292880955326191623218291923, −5.43656342125336552568203560689, −5.37482015543001329439013809427, −4.69406347426726014261216827288, −4.51309299834291170978293787008, −3.79238502715277947154250337415, −3.33531280816128776210715032117, −2.41621043666811926183211528055, −2.15755151579750822439488295890, −2.01210570818027583291539427535, −1.39601899629840364230876764432, 0, 0,
1.39601899629840364230876764432, 2.01210570818027583291539427535, 2.15755151579750822439488295890, 2.41621043666811926183211528055, 3.33531280816128776210715032117, 3.79238502715277947154250337415, 4.51309299834291170978293787008, 4.69406347426726014261216827288, 5.37482015543001329439013809427, 5.43656342125336552568203560689, 5.88292880955326191623218291923, 6.51428847088952488083938286690, 6.78172647251795419502247490688, 6.94452044853434994534463211027, 7.73379965291232005433046568282, 8.168851354407513143474785003809, 8.458460268400165322467861212245, 9.195253064873447991330262613729
