L(s) = 1 | + 2-s + (−1.67 − 0.425i)3-s + 4-s + (2.18 + 0.461i)5-s + (−1.67 − 0.425i)6-s + 8-s + (2.63 + 1.42i)9-s + (2.18 + 0.461i)10-s − 4.08i·11-s + (−1.67 − 0.425i)12-s − 3.50·13-s + (−3.47 − 1.70i)15-s + 16-s + 0.437i·17-s + (2.63 + 1.42i)18-s − 7.69i·19-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.969 − 0.245i)3-s + 0.5·4-s + (0.978 + 0.206i)5-s + (−0.685 − 0.173i)6-s + 0.353·8-s + (0.879 + 0.475i)9-s + (0.691 + 0.145i)10-s − 1.23i·11-s + (−0.484 − 0.122i)12-s − 0.970·13-s + (−0.897 − 0.440i)15-s + 0.250·16-s + 0.106i·17-s + (0.621 + 0.336i)18-s − 1.76i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.639 + 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.639 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.235100119\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.235100119\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (1.67 + 0.425i)T \) |
| 5 | \( 1 + (-2.18 - 0.461i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 4.08iT - 11T^{2} \) |
| 13 | \( 1 + 3.50T + 13T^{2} \) |
| 17 | \( 1 - 0.437iT - 17T^{2} \) |
| 19 | \( 1 + 7.69iT - 19T^{2} \) |
| 23 | \( 1 - 7.39T + 23T^{2} \) |
| 29 | \( 1 + 4.95iT - 29T^{2} \) |
| 31 | \( 1 - 6.81iT - 31T^{2} \) |
| 37 | \( 1 - 4.51iT - 37T^{2} \) |
| 41 | \( 1 - 1.34T + 41T^{2} \) |
| 43 | \( 1 + 6.67iT - 43T^{2} \) |
| 47 | \( 1 - 2.41iT - 47T^{2} \) |
| 53 | \( 1 - 0.424T + 53T^{2} \) |
| 59 | \( 1 - 2.75T + 59T^{2} \) |
| 61 | \( 1 + 5.75iT - 61T^{2} \) |
| 67 | \( 1 + 11.3iT - 67T^{2} \) |
| 71 | \( 1 - 12.1iT - 71T^{2} \) |
| 73 | \( 1 - 9.06T + 73T^{2} \) |
| 79 | \( 1 - 3.37T + 79T^{2} \) |
| 83 | \( 1 + 17.1iT - 83T^{2} \) |
| 89 | \( 1 + 4.52T + 89T^{2} \) |
| 97 | \( 1 - 7.93T + 97T^{2} \) |
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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
−9.536373662298719501548372031975, −8.665809798833932559144843593412, −7.33648722034441218146211186725, −6.73766233926046764227576338461, −6.07627767784185094397780960684, −5.14717802566426540827866838031, −4.80089291603869089386978164817, −3.21496611237070067181117626435, −2.29887743096985291050328980231, −0.859921581255463906971822246500,
1.39127585039797886952106887815, 2.45124840416675328765969181116, 3.89933649657287004673433859713, 4.85424557284577886755679659070, 5.32762575693909694177679432911, 6.16345364234276967803762999367, 6.96422928502194265683906995583, 7.65497852810201549485589753214, 9.155059644300777055376672115249, 9.854286626323409133799957305755