L(s) = 1 | + (2 − 3.46i)2-s + (−7.99 − 13.8i)4-s + (20.8 + 36.0i)5-s + (101. − 176. i)7-s − 63.9·8-s + 166.·10-s + (235. − 407. i)11-s + (−241. − 418. i)13-s + (−406. − 704. i)14-s + (−128 + 221. i)16-s − 1.25e3·17-s + 1.97e3·19-s + (332. − 576. i)20-s + (−940. − 1.62e3i)22-s + (239. + 414. i)23-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.372 + 0.644i)5-s + (0.784 − 1.35i)7-s − 0.353·8-s + 0.526·10-s + (0.585 − 1.01i)11-s + (−0.396 − 0.686i)13-s + (−0.554 − 0.960i)14-s + (−0.125 + 0.216i)16-s − 1.05·17-s + 1.25·19-s + (0.186 − 0.322i)20-s + (−0.414 − 0.717i)22-s + (0.0942 + 0.163i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0177 + 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0177 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.49602 - 1.52283i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.49602 - 1.52283i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2 + 3.46i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-20.8 - 36.0i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 7 | \( 1 + (-101. + 176. i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-235. + 407. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (241. + 418. i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 + 1.25e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.97e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (-239. - 414. i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (580. - 1.00e3i)T + (-1.02e7 - 1.77e7i)T^{2} \) |
| 31 | \( 1 + (-1.18e3 - 2.05e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 - 8.18e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-8.75e3 - 1.51e4i)T + (-5.79e7 + 1.00e8i)T^{2} \) |
| 43 | \( 1 + (1.14e4 - 1.98e4i)T + (-7.35e7 - 1.27e8i)T^{2} \) |
| 47 | \( 1 + (8.68e3 - 1.50e4i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 - 5.39e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + (2.22e4 + 3.85e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (2.08e3 - 3.60e3i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (1.22e3 + 2.12e3i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + 2.18e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.03e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + (-2.52e4 + 4.37e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + (2.59e4 - 4.48e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 - 2.01e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (4.02e4 - 6.96e4i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−14.01806079270398748107322914004, −13.14921076301728949283136027235, −11.42096089294141322236293671300, −10.83043202382719744689157915034, −9.649955327722189662168692410381, −7.896115601941165256036092242544, −6.41266054615597715118500828673, −4.67365731859264296372406252038, −3.11828557792111657709099390349, −1.06317331775616167608535234690,
2.04268684842429992877001157765, 4.56226860576090330867967049325, 5.58125764953593959094938035481, 7.14485915373132959081428419479, 8.711473895047508031515137170859, 9.449349616822797478264959130797, 11.61810008474262769578267397388, 12.35268965910926244305328206603, 13.62943433658153801900203024180, 14.80133531577249939733919007397