L(s) = 1 | + (−1.04 − 1.80i)2-s + (1.04 + 1.38i)3-s + (−1.17 + 2.03i)4-s + (1.41 − 3.32i)6-s + (2.04 + 3.53i)7-s + 0.734·8-s + (−0.824 + 2.88i)9-s + (0.675 + 1.17i)11-s + (−4.04 + 0.498i)12-s + (0.324 − 0.561i)13-s + (4.26 − 7.38i)14-s + (1.58 + 2.74i)16-s + 1.35·17-s + (6.07 − 1.52i)18-s + 0.648·19-s + ⋯ |
L(s) = 1 | + (−0.737 − 1.27i)2-s + (0.602 + 0.798i)3-s + (−0.587 + 1.01i)4-s + (0.575 − 1.35i)6-s + (0.772 + 1.33i)7-s + 0.259·8-s + (−0.274 + 0.961i)9-s + (0.203 + 0.353i)11-s + (−1.16 + 0.143i)12-s + (0.0898 − 0.155i)13-s + (1.13 − 1.97i)14-s + (0.396 + 0.686i)16-s + 0.327·17-s + (1.43 − 0.358i)18-s + 0.148·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.03833 - 0.0538967i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.03833 - 0.0538967i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.04 - 1.38i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (1.04 + 1.80i)T + (-1 + 1.73i)T^{2} \) |
| 7 | \( 1 + (-2.04 - 3.53i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.675 - 1.17i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.324 + 0.561i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 1.35T + 17T^{2} \) |
| 19 | \( 1 - 0.648T + 19T^{2} \) |
| 23 | \( 1 + (-2.39 + 4.14i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.93 + 3.35i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.84 + 6.66i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 7.52T + 37T^{2} \) |
| 41 | \( 1 + (-0.0898 + 0.155i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.410 + 0.710i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.45 - 9.44i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 4.17T + 53T^{2} \) |
| 59 | \( 1 + (2.08 - 3.61i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.91 - 3.30i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.07 + 7.05i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6.11T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 + (5.17 + 8.95i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (6.12 + 10.6i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 3T + 89T^{2} \) |
| 97 | \( 1 + (6.79 + 11.7i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.89801010369161024984703932710, −11.17361607235606867070251841276, −10.25528515158058000645720275407, −9.361422803280399428420053456174, −8.723349194627634555232788543288, −7.900288131552896540614140708207, −5.77441286458787245560042476342, −4.46184693724550207779768763123, −2.98748582797083211699318964800, −2.00878147987189130033841711536,
1.18493927446267079811972016302, 3.53067851786585736663135382453, 5.30449963151017578807392288001, 6.74085300548243412512766178579, 7.25708199336196913567605157015, 8.129451460096030230929308998960, 8.850885898276287683871225185260, 9.983837455894586225769554544699, 11.21806280202796792631008276844, 12.37231623556755952884999856927