| L(s) = 1 | − i·2-s + 3.10·3-s − 4-s − 3.10i·6-s + 4.10i·7-s + i·8-s + 6.64·9-s + 1.53i·11-s − 3.10·12-s + (1.26 − 3.37i)13-s + 4.10·14-s + 16-s − 3·17-s − 6.64i·18-s + 1.10i·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 1.79·3-s − 0.5·4-s − 1.26i·6-s + 1.55i·7-s + 0.353i·8-s + 2.21·9-s + 0.463i·11-s − 0.896·12-s + (0.352 − 0.935i)13-s + 1.09·14-s + 0.250·16-s − 0.727·17-s − 1.56i·18-s + 0.253i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.55126 - 0.463912i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.55126 - 0.463912i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-1.26 + 3.37i)T \) |
| good | 3 | \( 1 - 3.10T + 3T^{2} \) |
| 7 | \( 1 - 4.10iT - 7T^{2} \) |
| 11 | \( 1 - 1.53iT - 11T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 - 1.10iT - 19T^{2} \) |
| 23 | \( 1 - 6.74T + 23T^{2} \) |
| 29 | \( 1 + 5.56T + 29T^{2} \) |
| 31 | \( 1 + 8.64iT - 31T^{2} \) |
| 37 | \( 1 + 4.53iT - 37T^{2} \) |
| 41 | \( 1 - 7.85iT - 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 10.8iT - 47T^{2} \) |
| 53 | \( 1 - 0.433T + 53T^{2} \) |
| 59 | \( 1 - 13.7iT - 59T^{2} \) |
| 61 | \( 1 + 14.3T + 61T^{2} \) |
| 67 | \( 1 + 3.74iT - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 10.7iT - 73T^{2} \) |
| 79 | \( 1 + 6.53T + 79T^{2} \) |
| 83 | \( 1 + 4.46iT - 83T^{2} \) |
| 89 | \( 1 - 1.85iT - 89T^{2} \) |
| 97 | \( 1 + 3.78iT - 97T^{2} \) |
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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
−10.28527415207088114855918391313, −9.323691971612487570592697382015, −8.978839369427310930496203380290, −8.215258544306347080229275839486, −7.36727778607536040636032733326, −5.86715463828762110512067674868, −4.65586493347089663291380378131, −3.44622670840664934619756605480, −2.65538604944242468917406741472, −1.86079623380938089510873369678,
1.44941193974799907672197631897, 3.13381882056046828511924879407, 3.95418395032328223186684306039, 4.78012498880481843033926573715, 6.68144943437631381833526328886, 7.13545804762807508546279364581, 7.968701296766738513207046052604, 8.861100495342501847245676653385, 9.302141379031426017432279799478, 10.37499734592380391771454711888
