| L(s) = 1 | + 2·2-s − 4·4-s + 16·7-s − 24·8-s + 26·9-s + 32·14-s − 16·16-s + 28·17-s + 52·18-s + 304·23-s − 64·28-s + 448·31-s + 160·32-s + 56·34-s − 104·36-s − 140·41-s + 608·46-s − 672·47-s − 494·49-s − 384·56-s + 896·62-s + 416·63-s + 448·64-s − 112·68-s − 144·71-s − 624·72-s + 588·73-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.863·7-s − 1.06·8-s + 0.962·9-s + 0.610·14-s − 1/4·16-s + 0.399·17-s + 0.680·18-s + 2.75·23-s − 0.431·28-s + 2.59·31-s + 0.883·32-s + 0.282·34-s − 0.481·36-s − 0.533·41-s + 1.94·46-s − 2.08·47-s − 1.44·49-s − 0.916·56-s + 1.83·62-s + 0.831·63-s + 7/8·64-s − 0.199·68-s − 0.240·71-s − 1.02·72-s + 0.942·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.607167402\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.607167402\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p^{3} T^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 26 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 8 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 2410 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 1594 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 14 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 12346 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 152 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 23578 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 224 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 42058 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 70 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 33878 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 336 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 296746 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 125130 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 444890 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 571034 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 72 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 294 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 464 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 846522 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 266 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 994 T + p^{3} T^{2} )^{2} \) |
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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
−12.31042408602567291887373935964, −11.90015358096315189946916544697, −11.20826438966790886470103959331, −11.08060340572120172874678933386, −10.19025919520793798687920447140, −9.747677830036982813410036887934, −9.461740363776981184786755855253, −8.518527378140111015622430427264, −8.399307168269159522179731553538, −7.77038059325047720517170938224, −6.80941709861388532632302775140, −6.77572819989509817789317106527, −5.81546792087560918706530723745, −5.11336679004699689316274306639, −4.62915306923212696174679123250, −4.47297858486166721844959550730, −3.29395723250374193763858938729, −2.94428432024269271217286686958, −1.58641848631677491843740680556, −0.817983316330969575452015451132,
0.817983316330969575452015451132, 1.58641848631677491843740680556, 2.94428432024269271217286686958, 3.29395723250374193763858938729, 4.47297858486166721844959550730, 4.62915306923212696174679123250, 5.11336679004699689316274306639, 5.81546792087560918706530723745, 6.77572819989509817789317106527, 6.80941709861388532632302775140, 7.77038059325047720517170938224, 8.399307168269159522179731553538, 8.518527378140111015622430427264, 9.461740363776981184786755855253, 9.747677830036982813410036887934, 10.19025919520793798687920447140, 11.08060340572120172874678933386, 11.20826438966790886470103959331, 11.90015358096315189946916544697, 12.31042408602567291887373935964
