| L(s) = 1 | + (−0.733 + 0.680i)2-s + (−2.81 + 0.424i)3-s + (0.0747 − 0.997i)4-s + (0.904 − 2.30i)5-s + (1.77 − 2.22i)6-s + (−1.30 − 2.29i)7-s + (0.623 + 0.781i)8-s + (4.88 − 1.50i)9-s + (0.904 + 2.30i)10-s + (−1.71 − 0.528i)11-s + (0.212 + 2.83i)12-s + (−1.16 − 5.12i)13-s + (2.52 + 0.795i)14-s + (−1.56 + 6.87i)15-s + (−0.988 − 0.149i)16-s + (−5.15 + 3.51i)17-s + ⋯ |
| L(s) = 1 | + (−0.518 + 0.480i)2-s + (−1.62 + 0.245i)3-s + (0.0373 − 0.498i)4-s + (0.404 − 1.03i)5-s + (0.724 − 0.908i)6-s + (−0.494 − 0.869i)7-s + (0.220 + 0.276i)8-s + (1.62 − 0.501i)9-s + (0.286 + 0.728i)10-s + (−0.517 − 0.159i)11-s + (0.0614 + 0.819i)12-s + (−0.324 − 1.42i)13-s + (0.674 + 0.212i)14-s + (−0.405 + 1.77i)15-s + (−0.247 − 0.0372i)16-s + (−1.24 + 0.852i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.104 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.104 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.269041 - 0.242205i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.269041 - 0.242205i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.733 - 0.680i)T \) |
| 7 | \( 1 + (1.30 + 2.29i)T \) |
| good | 3 | \( 1 + (2.81 - 0.424i)T + (2.86 - 0.884i)T^{2} \) |
| 5 | \( 1 + (-0.904 + 2.30i)T + (-3.66 - 3.40i)T^{2} \) |
| 11 | \( 1 + (1.71 + 0.528i)T + (9.08 + 6.19i)T^{2} \) |
| 13 | \( 1 + (1.16 + 5.12i)T + (-11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (5.15 - 3.51i)T + (6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (0.588 + 1.01i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.87 - 1.96i)T + (8.40 + 21.4i)T^{2} \) |
| 29 | \( 1 + (-0.395 - 0.190i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (-5.26 + 9.11i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.0500 + 0.667i)T + (-36.5 + 5.51i)T^{2} \) |
| 41 | \( 1 + (-3.47 - 4.35i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (-0.256 + 0.322i)T + (-9.56 - 41.9i)T^{2} \) |
| 47 | \( 1 + (4.51 - 4.18i)T + (3.51 - 46.8i)T^{2} \) |
| 53 | \( 1 + (-0.935 + 12.4i)T + (-52.4 - 7.89i)T^{2} \) |
| 59 | \( 1 + (-2.14 - 5.47i)T + (-43.2 + 40.1i)T^{2} \) |
| 61 | \( 1 + (0.314 + 4.20i)T + (-60.3 + 9.09i)T^{2} \) |
| 67 | \( 1 + (0.229 - 0.396i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-9.78 + 4.71i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (-7.71 - 7.15i)T + (5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (1.29 + 2.25i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.80 + 7.89i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (4.69 - 1.44i)T + (73.5 - 50.1i)T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−13.18060813995993238675155136693, −12.86427029474946293859730720227, −11.25647629868768117809457634517, −10.44811516587585152646607701347, −9.579043435207501801256012602218, −8.039416797417034736497676777545, −6.59362982819806014166687680791, −5.59838297478103755122325483771, −4.60318569468059179528868758002, −0.59793103291736296211181593991,
2.44113791447162738575537478226, 4.89347190740361084570648009204, 6.45233361069911631990512535261, 6.93904990815506697274981346988, 9.032656466679493529560493963854, 10.20166769688199585877195782975, 11.03775746089876716083523614232, 11.84023841308583811175524555626, 12.65852209548095092221237076781, 13.96523729080902987520662405003
