| L(s) = 1 | + (1.65 + 0.5i)3-s − 2·4-s + (2.5 + 1.65i)9-s + 3.31i·11-s + (−3.31 − i)12-s + 4·16-s + 9i·23-s + (3.31 + 4i)27-s − 5·31-s + (−1.65 + 5.5i)33-s + (−5 − 3.31i)36-s + 9.94·37-s − 6.63i·44-s + 12i·47-s + (6.63 + 2i)48-s + 7·49-s + ⋯ |
| L(s) = 1 | + (0.957 + 0.288i)3-s − 4-s + (0.833 + 0.552i)9-s + 1.00i·11-s + (−0.957 − 0.288i)12-s + 16-s + 1.87i·23-s + (0.638 + 0.769i)27-s − 0.898·31-s + (−0.288 + 0.957i)33-s + (−0.833 − 0.552i)36-s + 1.63·37-s − 1.00i·44-s + 1.75i·47-s + (0.957 + 0.288i)48-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.288 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.288 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.30111 + 0.966670i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30111 + 0.966670i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.65 - 0.5i)T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 3.31iT \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 9iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 - 9.94T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 3.31iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 9.94T + 67T^{2} \) |
| 71 | \( 1 + 16.5iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 16.5iT - 89T^{2} \) |
| 97 | \( 1 + 9.94T + 97T^{2} \) |
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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
−10.03572125079951896869994360833, −9.426823856934432194537065213390, −8.968912081457709747126649311255, −7.77675926821432098437620688173, −7.43181015916288493113809601077, −5.85178239071061506736739649709, −4.76763387295287100738563281793, −4.07750982349987772190396743637, −3.05999526166417652689461231315, −1.60823952968475727898726893515,
0.792718094811562857389913547233, 2.50011227736513341061658032683, 3.61731115066062440854324804012, 4.41297941920619156399297024038, 5.61078433429322471167381449637, 6.66616626958352135965378349550, 7.76563945612577702201482098201, 8.547735219873505716091301948908, 8.937916268462223111521524385905, 9.900258541993738320297909313044
