L(s) = 1 | + 1.03i·3-s + 2.44·5-s + i·7-s + 1.92·9-s + 5.06i·11-s − 2.54i·13-s + 2.53i·15-s + 5.30i·17-s − 1.14i·19-s − 1.03·21-s − 5.57·23-s + 0.980·25-s + 5.10i·27-s − 8.69i·29-s + 8.34·31-s + ⋯ |
L(s) = 1 | + 0.598i·3-s + 1.09·5-s + 0.377i·7-s + 0.641·9-s + 1.52i·11-s − 0.706i·13-s + 0.654i·15-s + 1.28i·17-s − 0.263i·19-s − 0.226·21-s − 1.16·23-s + 0.196·25-s + 0.982i·27-s − 1.61i·29-s + 1.49·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.236 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.236 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.041558257\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.041558257\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - iT \) |
| 41 | \( 1 + (6.22 + 1.51i)T \) |
good | 3 | \( 1 - 1.03iT - 3T^{2} \) |
| 5 | \( 1 - 2.44T + 5T^{2} \) |
| 11 | \( 1 - 5.06iT - 11T^{2} \) |
| 13 | \( 1 + 2.54iT - 13T^{2} \) |
| 17 | \( 1 - 5.30iT - 17T^{2} \) |
| 19 | \( 1 + 1.14iT - 19T^{2} \) |
| 23 | \( 1 + 5.57T + 23T^{2} \) |
| 29 | \( 1 + 8.69iT - 29T^{2} \) |
| 31 | \( 1 - 8.34T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 43 | \( 1 + 5.96T + 43T^{2} \) |
| 47 | \( 1 - 4.18iT - 47T^{2} \) |
| 53 | \( 1 - 6.93iT - 53T^{2} \) |
| 59 | \( 1 - 7.10T + 59T^{2} \) |
| 61 | \( 1 - 1.10T + 61T^{2} \) |
| 67 | \( 1 - 2.43iT - 67T^{2} \) |
| 71 | \( 1 + 6.81iT - 71T^{2} \) |
| 73 | \( 1 + 5.66T + 73T^{2} \) |
| 79 | \( 1 - 4.13iT - 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 - 2.55iT - 89T^{2} \) |
| 97 | \( 1 + 6.85iT - 97T^{2} \) |
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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
−10.08481828446526268008138361556, −9.541513435168822049182579588732, −8.380969113406536183663398096497, −7.57621807909568183763766266508, −6.40283840877648837268693981580, −5.81719595206811219141288400642, −4.71007016211480086268020275870, −4.06854915564118238881925113608, −2.53652084213567052678406128233, −1.65508093686480272992324535408,
0.950249308511799129717599226475, 2.04158344264765474592717744658, 3.21863486555721237280046503514, 4.47222705364282604627161005078, 5.54310489378634267865060972689, 6.38316001595563348718651208177, 6.94551557582497257814833300349, 8.004066934780759141001534866502, 8.777254669016198698985820653375, 9.827590561885530500167365147961