| L(s) = 1 | + 0.414i·2-s + i·3-s + 1.82·4-s − 0.414·6-s − 4.82i·7-s + 1.58i·8-s − 9-s − 11-s + 1.82i·12-s − 5.65i·13-s + 1.99·14-s + 3·16-s − 6.82i·17-s − 0.414i·18-s + 1.17·19-s + ⋯ |
| L(s) = 1 | + 0.292i·2-s + 0.577i·3-s + 0.914·4-s − 0.169·6-s − 1.82i·7-s + 0.560i·8-s − 0.333·9-s − 0.301·11-s + 0.527i·12-s − 1.56i·13-s + 0.534·14-s + 0.750·16-s − 1.65i·17-s − 0.0976i·18-s + 0.268·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.75759 - 0.414910i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.75759 - 0.414910i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 0.414iT - 2T^{2} \) |
| 7 | \( 1 + 4.82iT - 7T^{2} \) |
| 13 | \( 1 + 5.65iT - 13T^{2} \) |
| 17 | \( 1 + 6.82iT - 17T^{2} \) |
| 19 | \( 1 - 1.17T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 0.828T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 0.343iT - 37T^{2} \) |
| 41 | \( 1 + 0.828T + 41T^{2} \) |
| 43 | \( 1 - 3.17iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 - 13.3iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 0.343T + 61T^{2} \) |
| 67 | \( 1 - 5.65iT - 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 - 11.3iT - 73T^{2} \) |
| 79 | \( 1 - 8.48T + 79T^{2} \) |
| 83 | \( 1 - 10iT - 83T^{2} \) |
| 89 | \( 1 - 7.65T + 89T^{2} \) |
| 97 | \( 1 - 0.343iT - 97T^{2} \) |
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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
−10.22196916025729619598586871455, −9.630992226908692761741182085803, −8.151664135372379631135883524660, −7.48775495463780875543976542710, −6.95885925193177750885125328827, −5.69058803795240891829749169166, −4.90143341949170828001505059695, −3.63197644658420660360651121604, −2.80180300784318289124310698989, −0.885607331966827786438884359069,
1.84308860943031413382342096906, 2.30864727305991836298526744164, 3.58912204404934569321063156039, 5.13321722628245346220217454604, 6.25714136758179875364443566652, 6.49915623841898382295793606365, 7.79476598908460522041186935245, 8.599341278597317257799094183927, 9.339098454120590056275019911476, 10.43969702223941326611119859392
