L(s) = 1 | + 3·3-s + 6.58i·5-s + (15.1 + 10.5i)7-s + 9·9-s − 54.2i·11-s + 40.9i·13-s + 19.7i·15-s + 69.4i·17-s − 160.·19-s + (45.5 + 31.7i)21-s + 87.8i·23-s + 81.6·25-s + 27·27-s − 236.·29-s − 131.·31-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.589i·5-s + (0.820 + 0.571i)7-s + 0.333·9-s − 1.48i·11-s + 0.874i·13-s + 0.340i·15-s + 0.991i·17-s − 1.93·19-s + (0.473 + 0.330i)21-s + 0.796i·23-s + 0.652·25-s + 0.192·27-s − 1.51·29-s − 0.762·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.820 - 0.571i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.820 - 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.538571206\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.538571206\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 + (-15.1 - 10.5i)T \) |
good | 5 | \( 1 - 6.58iT - 125T^{2} \) |
| 11 | \( 1 + 54.2iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 40.9iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 69.4iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 160.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 87.8iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 236.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 131.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 23.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 112. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 194. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 269.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 120.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 338.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 267. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 275. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 270. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 1.23e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 691. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 430.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.22e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 381. iT - 9.12e5T^{2} \) |
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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
−9.322369585733411326502948238309, −8.634253391333340493127786041812, −8.210179426848016192800075748220, −7.16613390534826263350756400145, −6.25403625770143910961497062551, −5.53653831450275901785306553117, −4.25807276951648140331931508329, −3.50662416650347600072529323420, −2.36838051965825867150720877555, −1.55173698531274689653117922012,
0.29323056841764706620819588468, 1.63773188943924853646488162614, 2.44627995530951112986735261877, 3.88557211811057414382940491189, 4.62654548482810433982764526744, 5.25940229512426810567830852787, 6.69545856529276242841922356252, 7.40795985244579541968579647458, 8.126714723240908710016689816240, 8.844868033487025970790802776971