| L(s) = 1 | + 5.19·3-s − 29.7·5-s − 12.9i·7-s + 27·9-s + (100. + 67.4i)11-s + 36.6i·13-s − 154.·15-s − 464. i·17-s + 327. i·19-s − 67.2i·21-s − 396.·23-s + 259.·25-s + 140.·27-s + 1.15e3i·29-s − 437.·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.18·5-s − 0.264i·7-s + 0.333·9-s + (0.830 + 0.557i)11-s + 0.216i·13-s − 0.687·15-s − 1.60i·17-s + 0.907i·19-s − 0.152i·21-s − 0.750·23-s + 0.415·25-s + 0.192·27-s + 1.37i·29-s − 0.455·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 + 0.557i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.830 + 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.904222968\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.904222968\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 5.19T \) |
| 11 | \( 1 + (-100. - 67.4i)T \) |
| good | 5 | \( 1 + 29.7T + 625T^{2} \) |
| 7 | \( 1 + 12.9iT - 2.40e3T^{2} \) |
| 13 | \( 1 - 36.6iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 464. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 327. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 396.T + 2.79e5T^{2} \) |
| 29 | \( 1 - 1.15e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 437.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 276.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 2.78e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 1.66e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 1.85e3T + 4.87e6T^{2} \) |
| 53 | \( 1 - 3.74e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 6.57e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + 5.14e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 3.46e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 6.03e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 3.58e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 9.71e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 1.18e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 2.58e3T + 6.27e7T^{2} \) |
| 97 | \( 1 - 1.55e3T + 8.85e7T^{2} \) |
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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
−10.07575125825313221339726250705, −9.181142346755676958611730336305, −8.389654432938385410380062503061, −7.32386563932464871923034834118, −6.99762191350414139586864671803, −5.34507774261424057349424273850, −4.10189177503878156553185677919, −3.59746752929155843048957921255, −2.11336299606985553517634299869, −0.61408682942961136964102880626,
0.886817712932012517965164771715, 2.43355163907506626760155906545, 3.75398995812854343214628591111, 4.20451948876025094267885953312, 5.79144486614505241379522228289, 6.78519319190404957079462362031, 7.929737644925808914082910917312, 8.368017672643177410878660426062, 9.281082014257540418772207362823, 10.33669477105330223204718069259
