| L(s) = 1 | + 2.78i·5-s − 1.16i·7-s − 0.827i·11-s + (−2.92 + 2.10i)13-s − 0.0570·17-s + 8.44i·19-s − 2.36·23-s − 2.75·25-s − 5.34·29-s − 0.406i·31-s + 3.24·35-s − 5.42i·37-s − 8.70i·41-s − 10.2·43-s − 8.73i·47-s + ⋯ |
| L(s) = 1 | + 1.24i·5-s − 0.439i·7-s − 0.249i·11-s + (−0.811 + 0.584i)13-s − 0.0138·17-s + 1.93i·19-s − 0.492·23-s − 0.550·25-s − 0.991·29-s − 0.0729i·31-s + 0.547·35-s − 0.891i·37-s − 1.35i·41-s − 1.56·43-s − 1.27i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4212 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 + 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4212 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1459262227\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1459262227\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (2.92 - 2.10i)T \) |
| good | 5 | \( 1 - 2.78iT - 5T^{2} \) |
| 7 | \( 1 + 1.16iT - 7T^{2} \) |
| 11 | \( 1 + 0.827iT - 11T^{2} \) |
| 17 | \( 1 + 0.0570T + 17T^{2} \) |
| 19 | \( 1 - 8.44iT - 19T^{2} \) |
| 23 | \( 1 + 2.36T + 23T^{2} \) |
| 29 | \( 1 + 5.34T + 29T^{2} \) |
| 31 | \( 1 + 0.406iT - 31T^{2} \) |
| 37 | \( 1 + 5.42iT - 37T^{2} \) |
| 41 | \( 1 + 8.70iT - 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 + 8.73iT - 47T^{2} \) |
| 53 | \( 1 - 2.12T + 53T^{2} \) |
| 59 | \( 1 + 11.1iT - 59T^{2} \) |
| 61 | \( 1 - 0.450T + 61T^{2} \) |
| 67 | \( 1 - 9.27iT - 67T^{2} \) |
| 71 | \( 1 - 6.15iT - 71T^{2} \) |
| 73 | \( 1 - 8.45iT - 73T^{2} \) |
| 79 | \( 1 + 8.80T + 79T^{2} \) |
| 83 | \( 1 - 10.5iT - 83T^{2} \) |
| 89 | \( 1 + 5.24iT - 89T^{2} \) |
| 97 | \( 1 + 0.0879iT - 97T^{2} \) |
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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
−8.751857342476845925389444058636, −7.992652058269919212968106621329, −7.19796030330960902450390666796, −6.87697033067604195421871289283, −5.91707197391880673242953431956, −5.30031691911332795611873296749, −3.93901693734674395260298263352, −3.68185414061887857246074868987, −2.50965255909206421924710355532, −1.72664755998718339769090167165,
0.04081489454384834569708070829, 1.22152022468771671260014770472, 2.34977518711517788198351022610, 3.19625656464877207595769185708, 4.54321801697129296734631922608, 4.83142835111617708035522051870, 5.58176955102146462189656508612, 6.46839767389797208480738960367, 7.36115984579474209850841129007, 8.017250826782709988681104643941
