L(s) = 1 | + 1.93i·2-s − i·3-s − 1.73·4-s + i·5-s + 1.93·6-s + (−0.189 + 2.63i)7-s + 0.517i·8-s − 9-s − 1.93·10-s + (−3 − 1.41i)11-s + 1.73i·12-s − 4.24·13-s + (−5.09 − 0.366i)14-s + 15-s − 4.46·16-s + 2.44·17-s + ⋯ |
L(s) = 1 | + 1.36i·2-s − 0.577i·3-s − 0.866·4-s + 0.447i·5-s + 0.788·6-s + (−0.0716 + 0.997i)7-s + 0.183i·8-s − 0.333·9-s − 0.610·10-s + (−0.904 − 0.426i)11-s + 0.500i·12-s − 1.17·13-s + (−1.36 − 0.0978i)14-s + 0.258·15-s − 1.11·16-s + 0.594·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.490 + 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.490 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5447405259\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5447405259\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (0.189 - 2.63i)T \) |
| 11 | \( 1 + (3 + 1.41i)T \) |
good | 2 | \( 1 - 1.93iT - 2T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 - 2.44T + 17T^{2} \) |
| 19 | \( 1 - 3.86T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 3.86iT - 29T^{2} \) |
| 31 | \( 1 + 10.9iT - 31T^{2} \) |
| 37 | \( 1 + 7.46T + 37T^{2} \) |
| 41 | \( 1 + 1.79T + 41T^{2} \) |
| 43 | \( 1 - 2.44iT - 43T^{2} \) |
| 47 | \( 1 - 10.9iT - 47T^{2} \) |
| 53 | \( 1 - 3.46T + 53T^{2} \) |
| 59 | \( 1 - 0.928iT - 59T^{2} \) |
| 61 | \( 1 + 4.89T + 61T^{2} \) |
| 67 | \( 1 - 1.46T + 67T^{2} \) |
| 71 | \( 1 + 6.92T + 71T^{2} \) |
| 73 | \( 1 + 9.14T + 73T^{2} \) |
| 79 | \( 1 + 1.03iT - 79T^{2} \) |
| 83 | \( 1 + 1.41T + 83T^{2} \) |
| 89 | \( 1 - 8.92iT - 89T^{2} \) |
| 97 | \( 1 + 8.92iT - 97T^{2} \) |
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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
−10.09844975392046297024428758539, −9.279098507066643048375965585609, −8.327820645677816122051805777540, −7.66387054539433885811314776041, −7.20215837019195293258568639298, −6.05290835353560346149038974314, −5.66924965835284280466492297232, −4.81301046058698394869242071860, −3.05398958307099662672436718674, −2.19416110604795549995704169564,
0.21940827639317248113648308335, 1.69388079825782314267971138023, 2.91434320461044949806842033515, 3.74643336800440352300703695399, 4.69585439250010610969551118998, 5.34314726132618526253813267872, 6.94179806291978944735624389215, 7.65088707980228655820551306775, 8.747219468905672229035642459563, 9.751764320413761713311130606329