L(s) = 1 | + (12.6 − 12.6i)5-s + 13.8i·7-s + (1.54 − 1.54i)11-s + (32.7 + 32.7i)13-s − 18.6·17-s + (86.4 + 86.4i)19-s + 134. i·23-s − 194. i·25-s + (59.7 + 59.7i)29-s + 31.5·31-s + (175. + 175. i)35-s + (89.1 − 89.1i)37-s − 210. i·41-s + (−119. + 119. i)43-s − 182.·47-s + ⋯ |
L(s) = 1 | + (1.13 − 1.13i)5-s + 0.749i·7-s + (0.0424 − 0.0424i)11-s + (0.699 + 0.699i)13-s − 0.266·17-s + (1.04 + 1.04i)19-s + 1.21i·23-s − 1.55i·25-s + (0.382 + 0.382i)29-s + 0.182·31-s + (0.847 + 0.847i)35-s + (0.396 − 0.396i)37-s − 0.801i·41-s + (−0.423 + 0.423i)43-s − 0.567·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.167i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.985 - 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.613572189\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.613572189\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-12.6 + 12.6i)T - 125iT^{2} \) |
| 7 | \( 1 - 13.8iT - 343T^{2} \) |
| 11 | \( 1 + (-1.54 + 1.54i)T - 1.33e3iT^{2} \) |
| 13 | \( 1 + (-32.7 - 32.7i)T + 2.19e3iT^{2} \) |
| 17 | \( 1 + 18.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + (-86.4 - 86.4i)T + 6.85e3iT^{2} \) |
| 23 | \( 1 - 134. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (-59.7 - 59.7i)T + 2.43e4iT^{2} \) |
| 31 | \( 1 - 31.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-89.1 + 89.1i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 + 210. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (119. - 119. i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 + 182.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-26.1 + 26.1i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + (-441. + 441. i)T - 2.05e5iT^{2} \) |
| 61 | \( 1 + (174. + 174. i)T + 2.26e5iT^{2} \) |
| 67 | \( 1 + (91.7 + 91.7i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 + 348. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 299. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 943.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-313. - 313. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 - 1.41e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.51e3T + 9.12e5T^{2} \) |
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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
−10.07495279175674569370917035530, −9.344261653028409004120162130289, −8.834953509233601283603749157838, −7.86781443894760645273215170301, −6.46153669280001320693936500756, −5.64748316596818043891481161183, −5.01169986666696186542008009636, −3.63673465990328815133221628172, −2.07519297875702675573573797471, −1.19680559789134606055581613003,
0.912778749431110011884182015611, 2.43336924552341363781255351242, 3.31709989167626574448445144906, 4.71115026910632670975477121929, 5.91645928923034707637540740845, 6.63895907087296686185736741146, 7.41172852982769507892578827303, 8.582431526562880975366279354628, 9.667987100343764239845601424385, 10.35378996345574066533575212421