| L(s) = 1 | + 3·2-s + 4-s + 25·7-s − 21·8-s + 15·11-s − 20·13-s + 75·14-s − 71·16-s + 72·17-s + 2·19-s + 45·22-s + 114·23-s − 60·26-s + 25·28-s − 30·29-s + 101·31-s − 45·32-s + 216·34-s + 430·37-s + 6·38-s + 30·41-s − 110·43-s + 15·44-s + 342·46-s − 330·47-s + 282·49-s − 20·52-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 1/8·4-s + 1.34·7-s − 0.928·8-s + 0.411·11-s − 0.426·13-s + 1.43·14-s − 1.10·16-s + 1.02·17-s + 0.0241·19-s + 0.436·22-s + 1.03·23-s − 0.452·26-s + 0.168·28-s − 0.192·29-s + 0.585·31-s − 0.248·32-s + 1.08·34-s + 1.91·37-s + 0.0256·38-s + 0.114·41-s − 0.390·43-s + 0.0513·44-s + 1.09·46-s − 1.02·47-s + 0.822·49-s − 0.0533·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.780492981\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.780492981\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 3 T + p^{3} T^{2} \) |
| 7 | \( 1 - 25 T + p^{3} T^{2} \) |
| 11 | \( 1 - 15 T + p^{3} T^{2} \) |
| 13 | \( 1 + 20 T + p^{3} T^{2} \) |
| 17 | \( 1 - 72 T + p^{3} T^{2} \) |
| 19 | \( 1 - 2 T + p^{3} T^{2} \) |
| 23 | \( 1 - 114 T + p^{3} T^{2} \) |
| 29 | \( 1 + 30 T + p^{3} T^{2} \) |
| 31 | \( 1 - 101 T + p^{3} T^{2} \) |
| 37 | \( 1 - 430 T + p^{3} T^{2} \) |
| 41 | \( 1 - 30 T + p^{3} T^{2} \) |
| 43 | \( 1 + 110 T + p^{3} T^{2} \) |
| 47 | \( 1 + 330 T + p^{3} T^{2} \) |
| 53 | \( 1 - 621 T + p^{3} T^{2} \) |
| 59 | \( 1 - 660 T + p^{3} T^{2} \) |
| 61 | \( 1 + 376 T + p^{3} T^{2} \) |
| 67 | \( 1 - 250 T + p^{3} T^{2} \) |
| 71 | \( 1 - 360 T + p^{3} T^{2} \) |
| 73 | \( 1 + 785 T + p^{3} T^{2} \) |
| 79 | \( 1 - 488 T + p^{3} T^{2} \) |
| 83 | \( 1 - 489 T + p^{3} T^{2} \) |
| 89 | \( 1 - 450 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1105 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.15578527391514071578256178353, −9.201628556666857511656862842308, −8.302178030610384780257163390781, −7.43459425599532309853966659738, −6.26742341908293050137852210866, −5.26248036721900535630363900076, −4.70404199047052966931723357475, −3.71361792850736576998427500569, −2.52198665200794686658933395410, −1.03211900351115389901864813923,
1.03211900351115389901864813923, 2.52198665200794686658933395410, 3.71361792850736576998427500569, 4.70404199047052966931723357475, 5.26248036721900535630363900076, 6.26742341908293050137852210866, 7.43459425599532309853966659738, 8.302178030610384780257163390781, 9.201628556666857511656862842308, 10.15578527391514071578256178353
