| L(s) = 1 | + 6·3-s + 24·5-s + 27·9-s + 44·11-s − 26·13-s + 144·15-s + 164·17-s − 48·19-s − 8·23-s + 238·25-s + 108·27-s + 404·29-s − 40·31-s + 264·33-s − 100·37-s − 156·39-s + 200·41-s + 616·43-s + 648·45-s + 324·47-s − 630·49-s + 984·51-s − 164·53-s + 1.05e3·55-s − 288·57-s − 140·59-s + 628·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 2.14·5-s + 9-s + 1.20·11-s − 0.554·13-s + 2.47·15-s + 2.33·17-s − 0.579·19-s − 0.0725·23-s + 1.90·25-s + 0.769·27-s + 2.58·29-s − 0.231·31-s + 1.39·33-s − 0.444·37-s − 0.640·39-s + 0.761·41-s + 2.18·43-s + 2.14·45-s + 1.00·47-s − 1.83·49-s + 2.70·51-s − 0.425·53-s + 2.58·55-s − 0.669·57-s − 0.308·59-s + 1.31·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(11.05109732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.05109732\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 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
−10.27980730776400394067776636620, −9.770701323103420460431909870279, −9.614037503897536650535820916523, −9.296797259332597811529216426574, −8.764138241710498833210539088719, −8.392720086460778074207726139323, −7.74692744715487557177130212898, −7.49909351511723033663309175861, −6.58907235768349528196182927778, −6.56774233062903416074053924048, −5.79946069360731658159358075157, −5.64459754672947573197855165095, −4.83736363374318329539729612758, −4.40826343846619671187500119753, −3.61143156369636993235568242995, −3.15153143650535655539178554888, −2.40507717432568676248969787613, −2.18899398287364693255483577102, −1.15223979646295715452197509725, −1.14245332056406843797872160453,
1.14245332056406843797872160453, 1.15223979646295715452197509725, 2.18899398287364693255483577102, 2.40507717432568676248969787613, 3.15153143650535655539178554888, 3.61143156369636993235568242995, 4.40826343846619671187500119753, 4.83736363374318329539729612758, 5.64459754672947573197855165095, 5.79946069360731658159358075157, 6.56774233062903416074053924048, 6.58907235768349528196182927778, 7.49909351511723033663309175861, 7.74692744715487557177130212898, 8.392720086460778074207726139323, 8.764138241710498833210539088719, 9.296797259332597811529216426574, 9.614037503897536650535820916523, 9.770701323103420460431909870279, 10.27980730776400394067776636620
