| L(s) = 1 | + i·2-s + 4-s + 3i·8-s + 3·9-s − 11-s − 2i·13-s − 16-s + 6i·17-s + 3i·18-s + 4·19-s − i·22-s − 4i·23-s + 2·26-s − 6·29-s − 8·31-s + 5i·32-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.5·4-s + 1.06i·8-s + 9-s − 0.301·11-s − 0.554i·13-s − 0.250·16-s + 1.45i·17-s + 0.707i·18-s + 0.917·19-s − 0.213i·22-s − 0.834i·23-s + 0.392·26-s − 1.11·29-s − 1.43·31-s + 0.883i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.30962 + 0.809391i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30962 + 0.809391i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - iT - 2T^{2} \) |
| 3 | \( 1 - 3T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 16iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−12.20675355460443389363383181244, −10.90794687354454144642014721749, −10.35651288740213495530735293664, −9.037887197330262027216882065514, −7.86650490421446811802287575091, −7.24996889059642868313010814463, −6.14066830454167328996642206422, −5.19694025163837720473738220247, −3.66850955045543300252274607757, −1.93823042745280027039669861818,
1.48799635321176176224017572114, 2.93007012260954588702810304303, 4.19185147450923176219971115246, 5.59457055083573837498819553075, 7.08794643128684311937364276526, 7.48769381985499203074271236958, 9.350269247137370398911964447819, 9.792268267361740489889061113069, 11.00865143116840149642500161602, 11.56935558661056245702572763122
