L(s) = 1 | − 2-s + 4-s + (−1.5 + 2.59i)5-s + (−2 + 1.73i)7-s − 8-s + (1.5 − 2.59i)10-s + (−1.5 − 2.59i)11-s + (−2.5 − 4.33i)13-s + (2 − 1.73i)14-s + 16-s + (1.5 − 2.59i)17-s + (−2.5 − 4.33i)19-s + (−1.5 + 2.59i)20-s + (1.5 + 2.59i)22-s + (−1.5 + 2.59i)23-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + (−0.670 + 1.16i)5-s + (−0.755 + 0.654i)7-s − 0.353·8-s + (0.474 − 0.821i)10-s + (−0.452 − 0.783i)11-s + (−0.693 − 1.20i)13-s + (0.534 − 0.462i)14-s + 0.250·16-s + (0.363 − 0.630i)17-s + (−0.573 − 0.993i)19-s + (−0.335 + 0.580i)20-s + (0.319 + 0.553i)22-s + (−0.312 + 0.541i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 + 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2 - 1.73i)T \) |
good | 5 | \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.5 + 4.33i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + (-3.5 - 6.06i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (5.5 - 9.52i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (5.5 - 9.52i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + (-1.5 + 2.59i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-7.5 - 12.9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.88709375465246007266692109328, −10.12767765604517983346958768164, −9.193245987751058934679503617583, −8.093802168587464372059090740728, −7.32697019660382802746618633381, −6.41352881130203080659637104353, −5.31324951697940326793430517168, −3.28084061081759045993439417565, −2.74102370887630031957619182797, 0,
1.85724875136122169554801817291, 3.80959599410311598621617583258, 4.71425038901120794203073840691, 6.21773708928578797316902363844, 7.31790371965350425379938717046, 8.047058264937251934934417284852, 9.045628396700969285927040373530, 9.836127976125770262399389548979, 10.60375925062863853851680724064