L(s) = 1 | − 0.340·5-s + (1.09 + 2.40i)7-s + 0.671·11-s + (1.62 + 2.81i)13-s + (1.10 + 1.90i)17-s + (0.242 − 0.419i)19-s − 4.18·23-s − 4.88·25-s + (−0.478 + 0.829i)29-s + (−1.04 + 1.80i)31-s + (−0.373 − 0.818i)35-s + (4.81 − 8.34i)37-s + (3.90 + 6.75i)41-s + (−3.66 + 6.34i)43-s + (−1.34 − 2.33i)47-s + ⋯ |
L(s) = 1 | − 0.152·5-s + (0.414 + 0.909i)7-s + 0.202·11-s + (0.450 + 0.779i)13-s + (0.266 + 0.462i)17-s + (0.0555 − 0.0961i)19-s − 0.873·23-s − 0.976·25-s + (−0.0889 + 0.154i)29-s + (−0.187 + 0.323i)31-s + (−0.0631 − 0.138i)35-s + (0.791 − 1.37i)37-s + (0.609 + 1.05i)41-s + (−0.558 + 0.967i)43-s + (−0.196 − 0.340i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0170 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0170 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.476242901\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.476242901\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.09 - 2.40i)T \) |
good | 5 | \( 1 + 0.340T + 5T^{2} \) |
| 11 | \( 1 - 0.671T + 11T^{2} \) |
| 13 | \( 1 + (-1.62 - 2.81i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.10 - 1.90i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.242 + 0.419i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 4.18T + 23T^{2} \) |
| 29 | \( 1 + (0.478 - 0.829i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.04 - 1.80i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.81 + 8.34i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.90 - 6.75i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.66 - 6.34i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.34 + 2.33i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.12 - 10.6i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.47 - 4.28i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.76 - 3.04i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.16 - 10.6i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 5.57T + 71T^{2} \) |
| 73 | \( 1 + (3.71 + 6.43i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.00 - 8.67i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.47 - 4.28i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-8.52 + 14.7i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.23 + 7.33i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.509702949261885060176594540444, −8.901879478102356620628263506834, −8.114833273264191682895018981773, −7.38891842580533998510843917457, −6.19730998778823014118066190903, −5.75502795386766464025005137520, −4.57291348926205249846513497626, −3.79201586231534730970805438904, −2.52001914948401173738437665974, −1.50008636649885447876029182677,
0.60121928372327693278321580107, 1.94188502271577576933241872126, 3.37780549005366838910196048415, 4.08604292265412577585617835199, 5.09709296134337196284699289252, 6.00002942310348902958572489395, 6.92519275348709277603650692583, 7.84904259313979089900668969080, 8.192446300424594041998030065743, 9.391713410848573059192117071989