L(s) = 1 | + (1.36 + 0.363i)2-s − 0.460·3-s + (1.73 + 0.993i)4-s + (2.19 + 0.423i)5-s + (−0.629 − 0.167i)6-s + i·7-s + (2.01 + 1.98i)8-s − 2.78·9-s + (2.84 + 1.37i)10-s − 4.70i·11-s + (−0.799 − 0.457i)12-s − 0.906·13-s + (−0.363 + 1.36i)14-s + (−1.01 − 0.195i)15-s + (2.02 + 3.44i)16-s + 7.25i·17-s + ⋯ |
L(s) = 1 | + (0.966 + 0.257i)2-s − 0.266·3-s + (0.867 + 0.496i)4-s + (0.981 + 0.189i)5-s + (−0.257 − 0.0683i)6-s + 0.377i·7-s + (0.710 + 0.703i)8-s − 0.929·9-s + (0.900 + 0.435i)10-s − 1.41i·11-s + (−0.230 − 0.132i)12-s − 0.251·13-s + (−0.0971 + 0.365i)14-s + (−0.261 − 0.0504i)15-s + (0.506 + 0.862i)16-s + 1.75i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.825 - 0.564i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.825 - 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.14007 + 0.662266i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.14007 + 0.662266i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.36 - 0.363i)T \) |
| 5 | \( 1 + (-2.19 - 0.423i)T \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 + 0.460T + 3T^{2} \) |
| 11 | \( 1 + 4.70iT - 11T^{2} \) |
| 13 | \( 1 + 0.906T + 13T^{2} \) |
| 17 | \( 1 - 7.25iT - 17T^{2} \) |
| 19 | \( 1 + 2.15iT - 19T^{2} \) |
| 23 | \( 1 + 5.89iT - 23T^{2} \) |
| 29 | \( 1 + 6.91iT - 29T^{2} \) |
| 31 | \( 1 + 8.02T + 31T^{2} \) |
| 37 | \( 1 - 6.21T + 37T^{2} \) |
| 41 | \( 1 + 6.21T + 41T^{2} \) |
| 43 | \( 1 + 9.32T + 43T^{2} \) |
| 47 | \( 1 + 3.80iT - 47T^{2} \) |
| 53 | \( 1 + 4.96T + 53T^{2} \) |
| 59 | \( 1 - 11.7iT - 59T^{2} \) |
| 61 | \( 1 + 6.20iT - 61T^{2} \) |
| 67 | \( 1 + 5.42T + 67T^{2} \) |
| 71 | \( 1 - 1.74T + 71T^{2} \) |
| 73 | \( 1 + 5.82iT - 73T^{2} \) |
| 79 | \( 1 - 8.05T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 11.5T + 89T^{2} \) |
| 97 | \( 1 - 11.6iT - 97T^{2} \) |
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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
−12.03080823006879934208601476382, −11.09487904160850583862647252265, −10.45996559139258373104900661404, −8.885867180458330998769481701705, −8.099920192015554285331266796653, −6.36166724954010064514747990228, −6.05183775182762331415071890128, −5.07804767143668233671888459792, −3.44907574685471904441651989885, −2.26758009077840959541880556883,
1.81310145682576998400054221730, 3.16374081426880783385381407799, 4.86860237279288029875575599873, 5.39039854084576835885574159019, 6.64446471355142773009391272332, 7.49027460134455389878828026557, 9.319638903394511248152759356472, 9.953556654780453871630796633423, 11.03564862564932819956853769078, 11.88013781583082858896491575349