L(s) = 1 | + (−1.10 + 1.91i)2-s + (1.23 + 2.14i)3-s + (−1.44 − 2.51i)4-s + (−1.06 + 1.83i)5-s − 5.47·6-s + 1.98·8-s + (−1.56 + 2.70i)9-s + (−2.34 − 4.06i)10-s + (−2.39 − 4.14i)11-s + (3.58 − 6.21i)12-s − 13-s − 5.25·15-s + (0.697 − 1.20i)16-s + (−1.88 − 3.27i)17-s + (−3.45 − 5.98i)18-s + (−1.78 + 3.08i)19-s + ⋯ |
L(s) = 1 | + (−0.782 + 1.35i)2-s + (0.714 + 1.23i)3-s + (−0.724 − 1.25i)4-s + (−0.474 + 0.822i)5-s − 2.23·6-s + 0.703·8-s + (−0.520 + 0.901i)9-s + (−0.742 − 1.28i)10-s + (−0.721 − 1.25i)11-s + (1.03 − 1.79i)12-s − 0.277·13-s − 1.35·15-s + (0.174 − 0.301i)16-s + (−0.458 − 0.793i)17-s + (−0.814 − 1.41i)18-s + (−0.409 + 0.708i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.367994 - 0.244783i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.367994 - 0.244783i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 + (1.10 - 1.91i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.23 - 2.14i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (1.06 - 1.83i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2.39 + 4.14i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (1.88 + 3.27i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.78 - 3.08i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.23 - 3.87i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 5.90T + 29T^{2} \) |
| 31 | \( 1 + (1.88 + 3.26i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.81 - 4.87i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 3.40T + 43T^{2} \) |
| 47 | \( 1 + (3.55 - 6.15i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.19 - 10.7i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.39 - 4.14i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.60 + 2.77i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.44 - 2.51i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 2.53T + 71T^{2} \) |
| 73 | \( 1 + (-3.85 - 6.66i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.58 + 4.48i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 3.46T + 83T^{2} \) |
| 89 | \( 1 + (-1.83 + 3.17i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 5.40T + 97T^{2} \) |
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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
−10.85041753516643497059480385657, −10.12055222714213318614032846841, −9.345781470475511552999918643003, −8.610395059913690472479325606381, −7.88003329474859377838926078968, −7.14125369192307231609143865047, −5.99899572158627130997233637636, −5.11028615211351340958752621347, −3.75230987967463537343016969537, −2.90293569261585238618466839908,
0.26997157794483499771257263499, 1.82330263511639808724171843942, 2.33110500640930476114746507996, 3.75313566394283012263812277780, 4.94786583144003691102395949110, 6.69842323198391815053758237848, 7.59086993502241630895377042209, 8.456124833636656614242142079755, 8.800354276732977025488071893893, 9.919922467113062221922776997097