| L(s) = 1 | + (−0.755 − 0.654i)2-s + (0.359 − 1.69i)3-s + (0.142 + 0.989i)4-s + (0.415 + 0.909i)5-s + (−1.38 + 1.04i)6-s + (0.816 + 2.78i)7-s + (0.540 − 0.841i)8-s + (−2.74 − 1.21i)9-s + (0.281 − 0.959i)10-s + (0.260 + 0.301i)11-s + (1.72 + 0.114i)12-s + (2.27 + 0.668i)13-s + (1.20 − 2.63i)14-s + (1.69 − 0.376i)15-s + (−0.959 + 0.281i)16-s + (−0.489 + 3.40i)17-s + ⋯ |
| L(s) = 1 | + (−0.534 − 0.463i)2-s + (0.207 − 0.978i)3-s + (0.0711 + 0.494i)4-s + (0.185 + 0.406i)5-s + (−0.563 + 0.426i)6-s + (0.308 + 1.05i)7-s + (0.191 − 0.297i)8-s + (−0.913 − 0.405i)9-s + (0.0890 − 0.303i)10-s + (0.0786 + 0.0908i)11-s + (0.498 + 0.0330i)12-s + (0.631 + 0.185i)13-s + (0.321 − 0.704i)14-s + (0.436 − 0.0973i)15-s + (−0.239 + 0.0704i)16-s + (−0.118 + 0.826i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.28564 - 0.209038i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.28564 - 0.209038i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.755 + 0.654i)T \) |
| 3 | \( 1 + (-0.359 + 1.69i)T \) |
| 5 | \( 1 + (-0.415 - 0.909i)T \) |
| 23 | \( 1 + (-4.78 + 0.260i)T \) |
| good | 7 | \( 1 + (-0.816 - 2.78i)T + (-5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (-0.260 - 0.301i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-2.27 - 0.668i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (0.489 - 3.40i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (0.0720 - 0.0103i)T + (18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-8.24 - 1.18i)T + (27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-7.97 - 5.12i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (0.619 + 0.282i)T + (24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (3.91 - 1.78i)T + (26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (2.36 + 3.68i)T + (-17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + 1.09iT - 47T^{2} \) |
| 53 | \( 1 + (7.85 - 2.30i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (-2.89 + 9.85i)T + (-49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-5.38 + 8.37i)T + (-25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (6.25 + 5.41i)T + (9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (-10.8 - 9.44i)T + (10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (1.74 + 12.1i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (0.274 - 0.935i)T + (-66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (7.05 - 15.4i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (-1.35 + 0.870i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (1.94 - 0.890i)T + (63.5 - 73.3i)T^{2} \) |
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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
−10.53315397982171059126264257045, −9.423688356255876299549056879444, −8.469749377608377682275224828099, −8.234937741996511644413670763002, −6.82038009197476158634431838733, −6.32632550339427878030001639478, −5.05541532857787404681296307641, −3.35319192896117161277259060210, −2.42714916192557037847989444671, −1.35758396948410355294822242045,
0.942152192059766477002304617998, 2.88523002878192576207151180334, 4.26432267799913803649550462127, 4.92705938285929511516895424289, 6.04946795879957389200354495132, 7.11178525429341096931216106513, 8.165969177052868627825668363519, 8.732748545550227608037591566045, 9.712465406111747527174229082368, 10.27615548826924328312029244449
