| L(s) = 1 | + (−0.415 + 0.909i)2-s + (0.959 + 0.281i)3-s + (−0.654 − 0.755i)4-s + (−0.841 + 0.540i)5-s + (−0.654 + 0.755i)6-s + (0.565 − 3.93i)7-s + (0.959 − 0.281i)8-s + (0.841 + 0.540i)9-s + (−0.142 − 0.989i)10-s + (0.826 + 1.80i)11-s + (−0.415 − 0.909i)12-s + (−0.745 − 5.18i)13-s + (3.34 + 2.14i)14-s + (−0.959 + 0.281i)15-s + (−0.142 + 0.989i)16-s + (−0.486 + 0.561i)17-s + ⋯ |
| L(s) = 1 | + (−0.293 + 0.643i)2-s + (0.553 + 0.162i)3-s + (−0.327 − 0.377i)4-s + (−0.376 + 0.241i)5-s + (−0.267 + 0.308i)6-s + (0.213 − 1.48i)7-s + (0.339 − 0.0996i)8-s + (0.280 + 0.180i)9-s + (−0.0450 − 0.313i)10-s + (0.249 + 0.545i)11-s + (−0.119 − 0.262i)12-s + (−0.206 − 1.43i)13-s + (0.893 + 0.574i)14-s + (−0.247 + 0.0727i)15-s + (−0.0355 + 0.247i)16-s + (−0.118 + 0.136i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.385i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.922 + 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.31417 - 0.263242i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31417 - 0.263242i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.415 - 0.909i)T \) |
| 3 | \( 1 + (-0.959 - 0.281i)T \) |
| 5 | \( 1 + (0.841 - 0.540i)T \) |
| 23 | \( 1 + (0.0381 + 4.79i)T \) |
| good | 7 | \( 1 + (-0.565 + 3.93i)T + (-6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (-0.826 - 1.80i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (0.745 + 5.18i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (0.486 - 0.561i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (1.86 + 2.15i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (-6.71 + 7.74i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (-0.449 + 0.132i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (-3.89 - 2.50i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (5.61 - 3.60i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-8.69 - 2.55i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 - 6.95T + 47T^{2} \) |
| 53 | \( 1 + (-0.148 + 1.03i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (-0.178 - 1.24i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (9.96 - 2.92i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (-0.997 + 2.18i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (4.15 - 9.09i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (2.26 + 2.61i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (0.734 + 5.10i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (-4.51 - 2.90i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (-12.3 - 3.61i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (5.14 - 3.30i)T + (40.2 - 88.2i)T^{2} \) |
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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
−10.39057469392845462983237719881, −9.604509444438813909847982451372, −8.383282046227060189562892494646, −7.82462845298396724571589099316, −7.12302273062845830585328016847, −6.21841601079549959165425699183, −4.68267270173655184356656420007, −4.12542379883299421989908784919, −2.74285452452469129634907532182, −0.78329375163790810964654427599,
1.61069387888678035955868744183, 2.63411307612086015235467256347, 3.77249893663105201099960798488, 4.87102617580089859161541688232, 6.05307689103102408497234831362, 7.22010881158584973851377727568, 8.274782598549001426546914579162, 9.019415936919282916825696160513, 9.238399249175899958887204873947, 10.57262256109597888101808391215
