L(s) = 1 | + 3.28·3-s + (1.54 − 1.62i)5-s + 2.73i·7-s + 7.80·9-s + 2.28·11-s + 4.92i·13-s + (5.06 − 5.32i)15-s − 5.97·17-s − 0.377·19-s + 8.97i·21-s + (0.582 − 4.76i)23-s + (−0.249 − 4.99i)25-s + 15.7·27-s + 3.70·29-s + 2.89i·31-s + ⋯ |
L(s) = 1 | + 1.89·3-s + (0.689 − 0.724i)5-s + 1.03i·7-s + 2.60·9-s + 0.689·11-s + 1.36i·13-s + (1.30 − 1.37i)15-s − 1.44·17-s − 0.0866·19-s + 1.95i·21-s + (0.121 − 0.992i)23-s + (−0.0498 − 0.998i)25-s + 3.03·27-s + 0.687·29-s + 0.519i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.015520962\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.015520962\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.54 + 1.62i)T \) |
| 23 | \( 1 + (-0.582 + 4.76i)T \) |
good | 3 | \( 1 - 3.28T + 3T^{2} \) |
| 7 | \( 1 - 2.73iT - 7T^{2} \) |
| 11 | \( 1 - 2.28T + 11T^{2} \) |
| 13 | \( 1 - 4.92iT - 13T^{2} \) |
| 17 | \( 1 + 5.97T + 17T^{2} \) |
| 19 | \( 1 + 0.377T + 19T^{2} \) |
| 29 | \( 1 - 3.70T + 29T^{2} \) |
| 31 | \( 1 - 2.89iT - 31T^{2} \) |
| 37 | \( 1 + 7.23T + 37T^{2} \) |
| 41 | \( 1 - 0.451T + 41T^{2} \) |
| 43 | \( 1 + 0.359iT - 43T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 53 | \( 1 + 9.69T + 53T^{2} \) |
| 59 | \( 1 + 10.7iT - 59T^{2} \) |
| 61 | \( 1 - 6.73iT - 61T^{2} \) |
| 67 | \( 1 - 7.06iT - 67T^{2} \) |
| 71 | \( 1 + 7.34iT - 71T^{2} \) |
| 73 | \( 1 + 1.24iT - 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 + 15.1iT - 83T^{2} \) |
| 89 | \( 1 + 12.1iT - 89T^{2} \) |
| 97 | \( 1 - 4.34T + 97T^{2} \) |
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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
−8.963055769181297721706586837678, −8.827957265929743347074510508399, −8.122687478754071555172966465959, −6.79064154588722471416922415327, −6.42429719936983304703771780148, −4.83042403381036817948496663505, −4.34121014524683869750843451419, −3.16433102578464596853436811511, −2.12462086488879804693119055323, −1.73217643144929320788337010450,
1.39133787422346331345357561768, 2.38720918909344496511837608478, 3.28633369474807696111668036528, 3.85010744557487069155125337206, 4.93064827350289468841520438549, 6.41468381547037737231485939453, 7.01043532204365984954708981142, 7.73911682781223458978792971392, 8.413918992159544723607521360155, 9.303676671729810082112901219146