L(s) = 1 | + (1 − 1.73i)2-s + (−1.99 − 3.46i)4-s + (3 − 5.19i)5-s − 7.99·8-s + (−6 − 10.3i)10-s + (15 + 25.9i)11-s + 2·13-s + (−8 + 13.8i)16-s + (33 + 57.1i)17-s + (26 − 45.0i)19-s − 24·20-s + 60·22-s + (57 − 98.7i)23-s + (44.5 + 77.0i)25-s + (2 − 3.46i)26-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.268 − 0.464i)5-s − 0.353·8-s + (−0.189 − 0.328i)10-s + (0.411 + 0.712i)11-s + 0.0426·13-s + (−0.125 + 0.216i)16-s + (0.470 + 0.815i)17-s + (0.313 − 0.543i)19-s − 0.268·20-s + 0.581·22-s + (0.516 − 0.895i)23-s + (0.355 + 0.616i)25-s + (0.0150 − 0.0261i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.721833961\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.721833961\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 + 1.73i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-3 + 5.19i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-15 - 25.9i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 2T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-33 - 57.1i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-26 + 45.0i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-57 + 98.7i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 72T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-98 - 169. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-143 + 247. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 378T + 6.89e4T^{2} \) |
| 43 | \( 1 - 164T + 7.95e4T^{2} \) |
| 47 | \( 1 + (114 - 197. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (174 + 301. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (174 + 301. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-53 + 91.7i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (298 + 516. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 630T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-521 - 902. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-44 + 76.2i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 1.44e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-687 + 1.18e3i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 34T + 9.12e5T^{2} \) |
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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
−9.509378862932006278287501647299, −9.089030824223987748543971069676, −7.989156998121775375580681715883, −6.92435915486380404192361273043, −5.95511027388913056515955278179, −4.99242889282001835768028142554, −4.22643940687768288615656634835, −3.08441164076010918014194814156, −1.89033874185499424632568491923, −0.845934592315256702655663330449,
0.941005237095029963353314272245, 2.64899658337377096953648270649, 3.55392066360507094528170575125, 4.66425588011873060933917611903, 5.75134536372381511610656400731, 6.30646798360923233614867252891, 7.35057479006364634303717958904, 7.991330549981919406288469274755, 9.092074987881284227973494814722, 9.726797115736370527362931270699