| L(s) = 1 | − 5.59i·3-s − 33.0·5-s + 43.4·7-s + 49.6·9-s − 13.5·11-s − 132. i·13-s + 184. i·15-s + 368.·17-s + (−230. + 277. i)19-s − 243. i·21-s − 14.2·23-s + 464.·25-s − 731. i·27-s − 307. i·29-s − 273. i·31-s + ⋯ |
| L(s) = 1 | − 0.622i·3-s − 1.32·5-s + 0.886·7-s + 0.612·9-s − 0.111·11-s − 0.783i·13-s + 0.821i·15-s + 1.27·17-s + (−0.638 + 0.769i)19-s − 0.551i·21-s − 0.0268·23-s + 0.743·25-s − 1.00i·27-s − 0.365i·29-s − 0.284i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.638 + 0.769i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.244853702\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.244853702\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (230. - 277. i)T \) |
| good | 3 | \( 1 + 5.59iT - 81T^{2} \) |
| 5 | \( 1 + 33.0T + 625T^{2} \) |
| 7 | \( 1 - 43.4T + 2.40e3T^{2} \) |
| 11 | \( 1 + 13.5T + 1.46e4T^{2} \) |
| 13 | \( 1 + 132. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 368.T + 8.35e4T^{2} \) |
| 23 | \( 1 + 14.2T + 2.79e5T^{2} \) |
| 29 | \( 1 + 307. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 273. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.92e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 1.39e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 2.11e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 1.45e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + 4.05e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 5.22e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 2.74e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 3.42e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 102. iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 3.64e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 7.13e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 2.01e3T + 4.74e7T^{2} \) |
| 89 | \( 1 - 7.51e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.75e4iT - 8.85e7T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.86623669572159244424691822248, −9.968057840112934597820719678212, −8.357862637142180806407044930526, −7.85424514265222058596729586770, −7.19620805441003264111911085420, −5.74955030480394082341983474436, −4.49004989370622574877148422458, −3.46938110190161043234239744484, −1.73297542811278471945651094472, −0.41716438205171651993591694013,
1.37699169812107880353188465769, 3.29637576425547281935731213739, 4.36865589314597336424070178563, 4.97329327181692140177960401911, 6.72383262381709465199714919748, 7.71873000710224186231897005666, 8.437599917811026601798339836268, 9.591624615078019957389451966631, 10.59461862359300189631352576271, 11.43829141175225041031968230288
