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 | \( 1 - 1.75e4iT - 8.85e7T^{2} \) |
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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
−11.43829141175225041031968230288, −10.59461862359300189631352576271, −9.591624615078019957389451966631, −8.437599917811026601798339836268, −7.71873000710224186231897005666, −6.72383262381709465199714919748, −4.97329327181692140177960401911, −4.36865589314597336424070178563, −3.29637576425547281935731213739, −1.37699169812107880353188465769,
0.41716438205171651993591694013, 1.73297542811278471945651094472, 3.46938110190161043234239744484, 4.49004989370622574877148422458, 5.74955030480394082341983474436, 7.19620805441003264111911085420, 7.85424514265222058596729586770, 8.357862637142180806407044930526, 9.968057840112934597820719678212, 10.86623669572159244424691822248