L(s) = 1 | + (−1.35 − 3.76i)2-s + (−12.3 + 10.2i)4-s + 2.66·5-s − 88.8i·7-s + (55.2 + 32.3i)8-s + (−3.61 − 10.0i)10-s + 72.9i·11-s − 173.·13-s + (−334. + 120. i)14-s + (46.7 − 251. i)16-s − 423.·17-s + 153. i·19-s + (−32.7 + 27.2i)20-s + (274. − 99.1i)22-s + 723. i·23-s + ⋯ |
L(s) = 1 | + (−0.339 − 0.940i)2-s + (−0.768 + 0.639i)4-s + 0.106·5-s − 1.81i·7-s + (0.862 + 0.505i)8-s + (−0.0361 − 0.100i)10-s + 0.602i·11-s − 1.02·13-s + (−1.70 + 0.616i)14-s + (0.182 − 0.983i)16-s − 1.46·17-s + 0.424i·19-s + (−0.0818 + 0.0680i)20-s + (0.567 − 0.204i)22-s + 1.36i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.639 - 0.768i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.639 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.127482 + 0.271787i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.127482 + 0.271787i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.35 + 3.76i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 2.66T + 625T^{2} \) |
| 7 | \( 1 + 88.8iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 72.9iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 173.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 423.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 153. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 723. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 366.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 642. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 535.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 2.14e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 615. iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 330. iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 4.62e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 3.46e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 4.64e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 6.67e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 6.44e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 4.73e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 2.08e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 9.29e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 5.70e3T + 6.27e7T^{2} \) |
| 97 | \( 1 - 6.97e3T + 8.85e7T^{2} \) |
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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
−12.20099964591107890578050390895, −11.12250149567781120261284885611, −10.19407298357460758407028257078, −9.483887588484167655949820820104, −7.88206519632996341921552737479, −7.02463619371487229361843168616, −4.74977422036161001394435339382, −3.73691590202222030758066679580, −1.86256621160982315710344715938, −0.14354681859381375727616163243,
2.38321031305154646931017364456, 4.75580261610172361196528128125, 5.82381805014601863785296526215, 6.86346671767599495128439089294, 8.485311092612526808056998213742, 8.943211566468646649600575816031, 10.17856392219750353491850988230, 11.62730189971921769968517308274, 12.71779377391355892325717371068, 13.86066177551823994132962936765