| L(s) = 1 | + (0.997 − 0.0713i)2-s + (−0.212 + 0.977i)3-s + (0.989 − 0.142i)4-s + (−1.32 − 1.79i)5-s + (−0.142 + 0.989i)6-s + (1.34 − 2.46i)7-s + (0.977 − 0.212i)8-s + (−0.909 − 0.415i)9-s + (−1.45 − 1.70i)10-s + (2.90 − 2.51i)11-s + (−0.0713 + 0.997i)12-s + (−4.18 + 2.28i)13-s + (1.16 − 2.55i)14-s + (2.04 − 0.914i)15-s + (0.959 − 0.281i)16-s + (−2.22 − 2.97i)17-s + ⋯ |
| L(s) = 1 | + (0.705 − 0.0504i)2-s + (−0.122 + 0.564i)3-s + (0.494 − 0.0711i)4-s + (−0.593 − 0.804i)5-s + (−0.0580 + 0.404i)6-s + (0.508 − 0.932i)7-s + (0.345 − 0.0751i)8-s + (−0.303 − 0.138i)9-s + (−0.459 − 0.537i)10-s + (0.874 − 0.757i)11-s + (−0.0205 + 0.287i)12-s + (−1.15 + 0.633i)13-s + (0.311 − 0.683i)14-s + (0.526 − 0.236i)15-s + (0.239 − 0.0704i)16-s + (−0.539 − 0.720i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.476 + 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.476 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.68909 - 1.00536i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.68909 - 1.00536i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.997 + 0.0713i)T \) |
| 3 | \( 1 + (0.212 - 0.977i)T \) |
| 5 | \( 1 + (1.32 + 1.79i)T \) |
| 23 | \( 1 + (-3.76 - 2.96i)T \) |
| good | 7 | \( 1 + (-1.34 + 2.46i)T + (-3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (-2.90 + 2.51i)T + (1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (4.18 - 2.28i)T + (7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (2.22 + 2.97i)T + (-4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (1.08 + 7.56i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (-9.50 - 1.36i)T + (27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (1.98 + 1.27i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (3.81 + 10.2i)T + (-27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (-0.540 - 1.18i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-6.89 - 1.50i)T + (39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (-4.20 - 4.20i)T + 47iT^{2} \) |
| 53 | \( 1 + (-4.12 - 2.25i)T + (28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (2.89 - 9.85i)T + (-49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (3.27 - 5.09i)T + (-25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (-0.349 - 4.88i)T + (-66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (-1.96 + 2.27i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-1.14 - 0.860i)T + (20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (6.34 + 1.86i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (15.6 - 5.82i)T + (62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (-6.94 + 4.46i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (2.33 + 0.872i)T + (73.3 + 63.5i)T^{2} \) |
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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
−10.64154016949787066859932652481, −9.255273421006153343988286789589, −8.871091022852919433265682649521, −7.42842008531969451382081333987, −6.90878849528421508199826004474, −5.46269445723772923648423144802, −4.47285142634284170507377297011, −4.26152494806851058988742804591, −2.80775959552152920127002833669, −0.874786672249030832833484477483,
1.90332448519906539688870660844, 2.89705975269916935728362956972, 4.17240011451687270024229653377, 5.17397463146597059700957322555, 6.29560924890284082862166492700, 6.90193065266898074417641705947, 7.893634322507715952521943265414, 8.568247594657264914049427719006, 10.03509875682832356585280243833, 10.73916405609845136934040563122
