L(s) = 1 | + (−0.247 + 0.427i)2-s + (0.674 + 1.59i)3-s + (0.877 + 1.52i)4-s + (0.5 + 0.866i)5-s + (−0.849 − 0.105i)6-s + (2.14 − 3.71i)7-s − 1.85·8-s + (−2.08 + 2.15i)9-s − 0.494·10-s + (−2.24 + 3.88i)11-s + (−1.83 + 2.42i)12-s + (0.5 + 0.866i)13-s + (1.05 + 1.83i)14-s + (−1.04 + 1.38i)15-s + (−1.29 + 2.24i)16-s + 1.57·17-s + ⋯ |
L(s) = 1 | + (−0.174 + 0.302i)2-s + (0.389 + 0.920i)3-s + (0.438 + 0.760i)4-s + (0.223 + 0.387i)5-s + (−0.346 − 0.0429i)6-s + (0.809 − 1.40i)7-s − 0.656·8-s + (−0.696 + 0.717i)9-s − 0.156·10-s + (−0.676 + 1.17i)11-s + (−0.529 + 0.700i)12-s + (0.138 + 0.240i)13-s + (0.282 + 0.490i)14-s + (−0.269 + 0.356i)15-s + (−0.324 + 0.561i)16-s + 0.381·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.585 - 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.779695 + 1.52588i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.779695 + 1.52588i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.674 - 1.59i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (-0.5 - 0.866i)T \) |
good | 2 | \( 1 + (0.247 - 0.427i)T + (-1 - 1.73i)T^{2} \) |
| 7 | \( 1 + (-2.14 + 3.71i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.24 - 3.88i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 - 1.57T + 17T^{2} \) |
| 19 | \( 1 - 1.86T + 19T^{2} \) |
| 23 | \( 1 + (-2.62 - 4.55i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.375 + 0.649i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (5.10 + 8.83i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 2.78T + 37T^{2} \) |
| 41 | \( 1 + (-3.18 - 5.50i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.79 - 6.57i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.13 + 7.16i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 0.0752T + 53T^{2} \) |
| 59 | \( 1 + (5.72 + 9.90i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.98 + 10.3i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.44 - 11.1i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 - 9.37T + 73T^{2} \) |
| 79 | \( 1 + (-3.04 + 5.28i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.306 - 0.530i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 16.2T + 89T^{2} \) |
| 97 | \( 1 + (-7.49 + 12.9i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.14073958619367933424174022933, −9.984559061250960445029520947875, −9.465243733500902319851020119109, −7.995870314657939617479815599487, −7.70751090862240872363284596297, −6.82359514078878345445040972874, −5.32900669539309693430387823916, −4.28686029548123438850418176923, −3.44195170802660776608221335015, −2.13506426372550140385484782914,
1.01480494893843027980633457920, 2.21421014198268510835319625559, 3.03519936334191694044787628538, 5.36652077691987264875481675654, 5.60368913535721248415210153509, 6.71240945706453184568488134645, 7.962167416693124519131813617888, 8.722306935737297664629677258586, 9.215038063083320428599662474152, 10.62029970197243545353925578135