L(s) = 1 | + (−0.0597 − 1.73i)3-s + (−0.567 + 2.58i)7-s + (−2.99 + 0.206i)9-s + (−0.793 − 0.457i)11-s − 4.31i·13-s + (−0.268 + 0.465i)17-s + (1.12 − 0.651i)19-s + (4.50 + 0.827i)21-s + (6.91 − 3.99i)23-s + (0.537 + 5.16i)27-s + 3.46i·29-s + (−5.56 − 3.21i)31-s + (−0.745 + 1.40i)33-s + (−2.52 − 4.36i)37-s + (−7.46 + 0.257i)39-s + ⋯ |
L(s) = 1 | + (−0.0345 − 0.999i)3-s + (−0.214 + 0.976i)7-s + (−0.997 + 0.0689i)9-s + (−0.239 − 0.138i)11-s − 1.19i·13-s + (−0.0651 + 0.112i)17-s + (0.258 − 0.149i)19-s + (0.983 + 0.180i)21-s + (1.44 − 0.832i)23-s + (0.103 + 0.994i)27-s + 0.642i·29-s + (−0.998 − 0.576i)31-s + (−0.129 + 0.243i)33-s + (−0.414 − 0.718i)37-s + (−1.19 + 0.0412i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5580407384\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5580407384\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.0597 + 1.73i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.567 - 2.58i)T \) |
good | 11 | \( 1 + (0.793 + 0.457i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 4.31iT - 13T^{2} \) |
| 17 | \( 1 + (0.268 - 0.465i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.12 + 0.651i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.91 + 3.99i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 3.46iT - 29T^{2} \) |
| 31 | \( 1 + (5.56 + 3.21i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.52 + 4.36i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 9.89T + 41T^{2} \) |
| 43 | \( 1 + 8.22T + 43T^{2} \) |
| 47 | \( 1 + (2.17 + 3.75i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.70 + 0.984i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (7.15 - 12.3i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (8.38 - 4.84i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.05 + 10.4i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 0.943iT - 71T^{2} \) |
| 73 | \( 1 + (-8.85 - 5.11i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.71 + 9.90i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 9.35T + 83T^{2} \) |
| 89 | \( 1 + (-0.874 - 1.51i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 10.7iT - 97T^{2} \) |
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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
−8.662388453267296109340457730379, −7.958551373709241908050374456677, −7.12576320031366923502846207539, −6.40901102623235814392368045909, −5.50920662537876039105632124212, −5.06684627327282422426482024578, −3.32611147612202282009637853841, −2.74667675747106120807207428544, −1.62438644844968546992526522179, −0.19157543616040323530551403359,
1.57775505793011651540173111996, 3.11367043011817116287508833021, 3.73759607958977732283784741168, 4.70940935683813151944062722664, 5.22502512086001330790212240563, 6.47818474735198834574794135806, 7.05470994920111024642238909474, 8.034875061413603293874627074994, 8.903677987343122103116596335851, 9.630870474215025614946259601236