| L(s) = 1 | + (−0.877 + 0.479i)2-s + (0.997 − 0.0713i)3-s + (0.540 − 0.841i)4-s + (−2.21 + 0.326i)5-s + (−0.841 + 0.540i)6-s + (0.672 − 1.80i)7-s + (−0.0713 + 0.997i)8-s + (0.989 − 0.142i)9-s + (1.78 − 1.34i)10-s + (1.31 + 4.48i)11-s + (0.479 − 0.877i)12-s + (−3.13 + 1.16i)13-s + (0.274 + 1.90i)14-s + (−2.18 + 0.483i)15-s + (−0.415 − 0.909i)16-s + (5.05 + 1.09i)17-s + ⋯ |
| L(s) = 1 | + (−0.620 + 0.338i)2-s + (0.575 − 0.0411i)3-s + (0.270 − 0.420i)4-s + (−0.989 + 0.146i)5-s + (−0.343 + 0.220i)6-s + (0.254 − 0.681i)7-s + (−0.0252 + 0.352i)8-s + (0.329 − 0.0474i)9-s + (0.564 − 0.425i)10-s + (0.397 + 1.35i)11-s + (0.138 − 0.253i)12-s + (−0.868 + 0.324i)13-s + (0.0732 + 0.509i)14-s + (−0.563 + 0.124i)15-s + (−0.103 − 0.227i)16-s + (1.22 + 0.266i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.04772 + 0.518949i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04772 + 0.518949i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.877 - 0.479i)T \) |
| 3 | \( 1 + (-0.997 + 0.0713i)T \) |
| 5 | \( 1 + (2.21 - 0.326i)T \) |
| 23 | \( 1 + (-4.76 - 0.568i)T \) |
| good | 7 | \( 1 + (-0.672 + 1.80i)T + (-5.29 - 4.58i)T^{2} \) |
| 11 | \( 1 + (-1.31 - 4.48i)T + (-9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (3.13 - 1.16i)T + (9.82 - 8.51i)T^{2} \) |
| 17 | \( 1 + (-5.05 - 1.09i)T + (15.4 + 7.06i)T^{2} \) |
| 19 | \( 1 + (1.04 + 0.669i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-4.64 - 7.22i)T + (-12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (-0.562 - 0.649i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (-5.85 + 7.82i)T + (-10.4 - 35.5i)T^{2} \) |
| 41 | \( 1 + (0.209 - 1.45i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-0.844 - 11.8i)T + (-42.5 + 6.11i)T^{2} \) |
| 47 | \( 1 + (-3.86 + 3.86i)T - 47iT^{2} \) |
| 53 | \( 1 + (3.54 + 1.32i)T + (40.0 + 34.7i)T^{2} \) |
| 59 | \( 1 + (9.22 + 4.21i)T + (38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (0.718 - 0.622i)T + (8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-5.62 - 10.2i)T + (-36.2 + 56.3i)T^{2} \) |
| 71 | \( 1 + (4.97 + 1.46i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (0.322 + 1.48i)T + (-66.4 + 30.3i)T^{2} \) |
| 79 | \( 1 + (-0.824 + 1.80i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (5.47 + 4.09i)T + (23.3 + 79.6i)T^{2} \) |
| 89 | \( 1 + (11.7 - 13.5i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (-11.8 + 8.84i)T + (27.3 - 93.0i)T^{2} \) |
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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
−10.41489932224358910447936148413, −9.652279163583083441335197700406, −8.839410014409066805271727444155, −7.72979387651287862701253755195, −7.40886775823213461836166335510, −6.65720427391513474117567367635, −4.93896151623375110728363353177, −4.16310326931950911931204917332, −2.85927387578595458376568194384, −1.28142571708657408134521164907,
0.848941403298165831052944140044, 2.66968128612933395267765908356, 3.42231481047421140266687290031, 4.65453390830486304387278720682, 5.90450701523279481896025547457, 7.21273865382188211533083799178, 8.039528770983235999000228870262, 8.531897974965408209717567773568, 9.338625280870180097839926809549, 10.28239736339215837248716076092
