| L(s) = 1 | + (1.25 − 3.28i)2-s + (4.62 − 0.242i)3-s + (−6.20 − 5.59i)4-s + (−4.99 + 0.114i)5-s + (5.03 − 15.4i)6-s + (−1.41 − 6.85i)7-s + (−13.6 + 6.95i)8-s + (12.4 − 1.30i)9-s + (−5.92 + 16.5i)10-s + (−1.35 + 12.9i)11-s + (−30.1 − 24.3i)12-s + (19.1 + 3.02i)13-s + (−24.2 − 3.97i)14-s + (−23.1 + 1.74i)15-s + (2.13 + 20.2i)16-s + (12.3 − 19.0i)17-s + ⋯ |
| L(s) = 1 | + (0.629 − 1.64i)2-s + (1.54 − 0.0808i)3-s + (−1.55 − 1.39i)4-s + (−0.999 + 0.0229i)5-s + (0.839 − 2.58i)6-s + (−0.202 − 0.979i)7-s + (−1.70 + 0.868i)8-s + (1.38 − 0.145i)9-s + (−0.592 + 1.65i)10-s + (−0.123 + 1.17i)11-s + (−2.50 − 2.03i)12-s + (1.47 + 0.233i)13-s + (−1.73 − 0.284i)14-s + (−1.54 + 0.116i)15-s + (0.133 + 1.26i)16-s + (0.728 − 1.12i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 + 0.481i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.643443 - 2.50979i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.643443 - 2.50979i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (4.99 - 0.114i)T \) |
| 7 | \( 1 + (1.41 + 6.85i)T \) |
| good | 2 | \( 1 + (-1.25 + 3.28i)T + (-2.97 - 2.67i)T^{2} \) |
| 3 | \( 1 + (-4.62 + 0.242i)T + (8.95 - 0.940i)T^{2} \) |
| 11 | \( 1 + (1.35 - 12.9i)T + (-118. - 25.1i)T^{2} \) |
| 13 | \( 1 + (-19.1 - 3.02i)T + (160. + 52.2i)T^{2} \) |
| 17 | \( 1 + (-12.3 + 19.0i)T + (-117. - 264. i)T^{2} \) |
| 19 | \( 1 + (11.2 - 10.1i)T + (37.7 - 359. i)T^{2} \) |
| 23 | \( 1 + (-16.9 - 6.51i)T + (393. + 353. i)T^{2} \) |
| 29 | \( 1 + (-8.58 + 2.78i)T + (680. - 494. i)T^{2} \) |
| 31 | \( 1 + (8.84 - 1.88i)T + (877. - 390. i)T^{2} \) |
| 37 | \( 1 + (-19.2 - 15.5i)T + (284. + 1.33e3i)T^{2} \) |
| 41 | \( 1 + (59.3 - 43.1i)T + (519. - 1.59e3i)T^{2} \) |
| 43 | \( 1 + (-8.13 - 8.13i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-9.34 + 6.06i)T + (898. - 2.01e3i)T^{2} \) |
| 53 | \( 1 + (-21.0 + 1.10i)T + (2.79e3 - 293. i)T^{2} \) |
| 59 | \( 1 + (25.0 - 56.3i)T + (-2.32e3 - 2.58e3i)T^{2} \) |
| 61 | \( 1 + (11.4 - 5.08i)T + (2.48e3 - 2.76e3i)T^{2} \) |
| 67 | \( 1 + (0.328 - 0.505i)T + (-1.82e3 - 4.10e3i)T^{2} \) |
| 71 | \( 1 + (-10.8 - 33.2i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-92.2 + 74.7i)T + (1.10e3 - 5.21e3i)T^{2} \) |
| 79 | \( 1 + (-22.3 + 105. i)T + (-5.70e3 - 2.53e3i)T^{2} \) |
| 83 | \( 1 + (134. - 68.4i)T + (4.04e3 - 5.57e3i)T^{2} \) |
| 89 | \( 1 + (2.18 + 4.91i)T + (-5.30e3 + 5.88e3i)T^{2} \) |
| 97 | \( 1 + (65.5 + 33.4i)T + (5.53e3 + 7.61e3i)T^{2} \) |
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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
−12.16539239782810271509115829469, −11.15421315773192332655226950649, −10.17827289616989688491572129116, −9.317774535528495474450851062361, −8.179175974294674979313549018939, −7.11754495957384133722011046532, −4.58974338645294867548290409493, −3.75477781346280338922033355102, −2.98808743471710201816571465339, −1.35462979858834253535738390791,
3.21908374644613269340968552979, 3.92635340744524742231304833252, 5.55417643519494369649379864578, 6.67057182340946082307901966094, 8.040728816186818012553510459387, 8.462048910720631872609183400743, 8.954472943203855920262483372663, 10.95262002062594465277411849783, 12.55525816291445965626426135548, 13.25764592654796239358957514975
