L(s) = 1 | + (0.173 − 0.984i)2-s + (−0.766 − 0.642i)3-s + (−0.939 − 0.342i)4-s + (−0.766 + 0.642i)6-s + (−0.5 + 0.866i)8-s + (0.5 − 0.866i)11-s + (0.499 + 0.866i)12-s + (0.766 + 0.642i)16-s + (0.347 − 1.96i)17-s + (−0.766 − 0.642i)22-s + (0.939 − 0.342i)24-s + (0.766 − 0.642i)25-s + (−0.499 + 0.866i)27-s + (0.766 − 0.642i)32-s + (−0.939 + 0.342i)33-s + (−1.87 − 0.684i)34-s + ⋯ |
L(s) = 1 | + (0.173 − 0.984i)2-s + (−0.766 − 0.642i)3-s + (−0.939 − 0.342i)4-s + (−0.766 + 0.642i)6-s + (−0.5 + 0.866i)8-s + (0.5 − 0.866i)11-s + (0.499 + 0.866i)12-s + (0.766 + 0.642i)16-s + (0.347 − 1.96i)17-s + (−0.766 − 0.642i)22-s + (0.939 − 0.342i)24-s + (0.766 − 0.642i)25-s + (−0.499 + 0.866i)27-s + (0.766 − 0.642i)32-s + (−0.939 + 0.342i)33-s + (−1.87 − 0.684i)34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.854 - 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.854 - 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7269444608\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7269444608\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.173 + 0.984i)T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \) |
| 5 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 17 | \( 1 + (-0.347 + 1.96i)T + (-0.939 - 0.342i)T^{2} \) |
| 23 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 29 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \) |
| 43 | \( 1 + (1.87 - 0.684i)T + (0.766 - 0.642i)T^{2} \) |
| 47 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 53 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 59 | \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \) |
| 61 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 67 | \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \) |
| 71 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 73 | \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \) |
| 79 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (-1.53 + 1.28i)T + (0.173 - 0.984i)T^{2} \) |
| 97 | \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \) |
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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
−8.757902657083771298384394926780, −7.78032448048648459742306757115, −6.81537809074580631905196000088, −6.16807559125946579169488967316, −5.30486207217264703921569866028, −4.68709766483578484052480593346, −3.46728412778949314313551068734, −2.82071232895791343736159884846, −1.46458321187513046661434171785, −0.51616929634515816952066625186,
1.67243791021429230629917055859, 3.42724340133725705905199335827, 4.14757706686331138797609810281, 4.93481174757101640928586508179, 5.48899829599636189105423616546, 6.38036907238376135502301982330, 6.87967449818450164875526747622, 7.926970063342483149334594278206, 8.465931071345011375132992736094, 9.360824406946524616588809363687