| L(s) = 1 | − 23.8·5-s + (−396. − 816. i)7-s − 643. i·11-s + 6.87e3i·13-s − 1.82e4·17-s − 4.04e4i·19-s + 1.05e5i·23-s − 7.75e4·25-s − 2.12e5i·29-s + 1.03e5i·31-s + (9.44e3 + 1.94e4i)35-s + 4.98e5·37-s + 7.99e5·41-s − 3.76e5·43-s − 8.98e5·47-s + ⋯ |
| L(s) = 1 | − 0.0853·5-s + (−0.436 − 0.899i)7-s − 0.145i·11-s + 0.867i·13-s − 0.901·17-s − 1.35i·19-s + 1.80i·23-s − 0.992·25-s − 1.62i·29-s + 0.623i·31-s + (0.0372 + 0.0767i)35-s + 1.61·37-s + 1.81·41-s − 0.722·43-s − 1.26·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.482 - 0.875i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.482 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.215655367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.215655367\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (396. + 816. i)T \) |
| good | 5 | \( 1 + 23.8T + 7.81e4T^{2} \) |
| 11 | \( 1 + 643. iT - 1.94e7T^{2} \) |
| 13 | \( 1 - 6.87e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 1.82e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.04e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 - 1.05e5iT - 3.40e9T^{2} \) |
| 29 | \( 1 + 2.12e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 - 1.03e5iT - 2.75e10T^{2} \) |
| 37 | \( 1 - 4.98e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 7.99e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.76e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 8.98e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 5.94e5iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 1.39e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.59e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 - 1.30e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.67e6iT - 9.09e12T^{2} \) |
| 73 | \( 1 - 5.32e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 + 7.54e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.29e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 8.40e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 8.49e6iT - 8.07e13T^{2} \) |
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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
−11.22224917836783912549161438106, −9.825300096634084201356952314609, −9.259048950624211771337084853577, −7.899724509367188724352453624505, −7.01358512598321691757989560154, −6.08948221826235996882367275103, −4.60904680633163843845679773682, −3.74756517218459821126478861671, −2.33365259794421451711319109799, −0.870662445657680833962032754371,
0.34948596418701747174966883748, 2.00984202520380923238037466823, 3.09165326239257564195069009929, 4.41320702609433784414737417258, 5.69256439461205179424233941080, 6.46897684205952748521314249855, 7.83532399056705242678048740578, 8.664668058701808405953239799783, 9.683859414269975888194993609986, 10.58344182887129837509784462386
