| L(s) = 1 | + (−0.996 − 0.0871i)2-s + (−1.57 − 0.724i)3-s + (0.984 + 0.173i)4-s + (−1.77 − 1.35i)5-s + (1.50 + 0.859i)6-s + (−1.16 − 0.813i)7-s + (−0.965 − 0.258i)8-s + (1.94 + 2.28i)9-s + (1.65 + 1.50i)10-s + (0.290 − 0.797i)11-s + (−1.42 − 0.986i)12-s + (0.499 + 5.70i)13-s + (1.08 + 0.911i)14-s + (1.81 + 3.41i)15-s + (0.939 + 0.342i)16-s + (−6.26 + 1.67i)17-s + ⋯ |
| L(s) = 1 | + (−0.704 − 0.0616i)2-s + (−0.908 − 0.418i)3-s + (0.492 + 0.0868i)4-s + (−0.795 − 0.605i)5-s + (0.613 + 0.350i)6-s + (−0.439 − 0.307i)7-s + (−0.341 − 0.0915i)8-s + (0.649 + 0.760i)9-s + (0.523 + 0.475i)10-s + (0.0874 − 0.240i)11-s + (−0.410 − 0.284i)12-s + (0.138 + 1.58i)13-s + (0.290 + 0.243i)14-s + (0.469 + 0.882i)15-s + (0.234 + 0.0855i)16-s + (−1.51 + 0.407i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.261 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.261 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.118624 + 0.155004i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.118624 + 0.155004i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.996 + 0.0871i)T \) |
| 3 | \( 1 + (1.57 + 0.724i)T \) |
| 5 | \( 1 + (1.77 + 1.35i)T \) |
| good | 7 | \( 1 + (1.16 + 0.813i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (-0.290 + 0.797i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.499 - 5.70i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (6.26 - 1.67i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-0.472 + 0.272i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.49 - 3.56i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (2.18 - 1.83i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (1.43 - 8.12i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-0.0155 - 0.0581i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (6.57 - 7.84i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (3.71 + 7.96i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (-4.34 + 6.20i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (-3.24 + 3.24i)T - 53iT^{2} \) |
| 59 | \( 1 + (1.79 - 0.652i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (0.615 + 3.49i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-1.81 + 0.158i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (4.81 + 2.77i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.64 - 9.85i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (8.97 + 10.6i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (0.723 - 8.27i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (-4.25 - 7.36i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (14.1 - 6.59i)T + (62.3 - 74.3i)T^{2} \) |
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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
−11.84256296765482243214742987816, −11.39855515281610433373873489845, −10.48996916274568985734823047518, −9.204115953431742512327209748417, −8.454644106048887132678575822482, −7.06735273390997888727221374214, −6.67648171134134511655030009645, −5.09255394602839091752836558579, −3.87293219170186276705782288096, −1.59606794397248343715359110075,
0.21068054182867804029188010328, 2.89205632745706218031482284962, 4.33906093857566049926461238031, 5.78369621737701830627401182672, 6.72587141207594098815361123196, 7.65697117751101886706719000827, 8.867171871131005996572083020303, 9.920325881437538863881237982594, 10.77277887979953774530138885079, 11.33088995019284416201791279764
