| L(s) = 1 | + (−1 − 2i)5-s − i·7-s − 4i·13-s − 4i·17-s + 4·19-s − 8i·23-s + (−3 + 4i)25-s + 2·29-s + 8·31-s + (−2 + i)35-s − 8i·37-s − 6·41-s + 8i·43-s + 8i·47-s − 49-s + ⋯ |
| L(s) = 1 | + (−0.447 − 0.894i)5-s − 0.377i·7-s − 1.10i·13-s − 0.970i·17-s + 0.917·19-s − 1.66i·23-s + (−0.600 + 0.800i)25-s + 0.371·29-s + 1.43·31-s + (−0.338 + 0.169i)35-s − 1.31i·37-s − 0.937·41-s + 1.21i·43-s + 1.16i·47-s − 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.376087145\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.376087145\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1 + 2i)T \) |
| 7 | \( 1 + iT \) |
| good | 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 12iT - 97T^{2} \) |
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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
−8.004927826163413504461626533484, −7.36588654966705740772883764385, −6.51989235796676517086171637572, −5.64048081960499733395249917330, −4.85784972299691061759209253665, −4.40950496735073322761564817086, −3.32882669670933586504865335904, −2.60320842424738364566102191766, −1.13134470133353895023009717085, −0.42186979409264261169633077859,
1.38982622360503197166803205182, 2.36491754498944588633482530071, 3.33931811545019351658388259794, 3.88923421219365185261251826106, 4.88150307405365661103634051360, 5.70819487195774787531514877327, 6.56460471553477907644419021495, 6.97518535310932800012111555790, 7.85509086689708353634441395414, 8.415765287315487114649108116390
