L(s) = 1 | + (−0.755 − 0.654i)2-s + (−0.989 − 0.142i)3-s + (0.142 + 0.989i)4-s + (0.415 + 0.909i)5-s + (0.654 + 0.755i)6-s + (0.540 + 1.84i)7-s + (0.540 − 0.841i)8-s + (0.959 + 0.281i)9-s + (0.281 − 0.959i)10-s − i·12-s + (0.797 − 1.74i)14-s + (−0.281 − 0.959i)15-s + (−0.959 + 0.281i)16-s + (−0.540 − 0.841i)18-s + (−0.841 + 0.540i)20-s + (−0.273 − 1.89i)21-s + ⋯ |
L(s) = 1 | + (−0.755 − 0.654i)2-s + (−0.989 − 0.142i)3-s + (0.142 + 0.989i)4-s + (0.415 + 0.909i)5-s + (0.654 + 0.755i)6-s + (0.540 + 1.84i)7-s + (0.540 − 0.841i)8-s + (0.959 + 0.281i)9-s + (0.281 − 0.959i)10-s − i·12-s + (0.797 − 1.74i)14-s + (−0.281 − 0.959i)15-s + (−0.959 + 0.281i)16-s + (−0.540 − 0.841i)18-s + (−0.841 + 0.540i)20-s + (−0.273 − 1.89i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5774362052\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5774362052\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.755 + 0.654i)T \) |
| 3 | \( 1 + (0.989 + 0.142i)T \) |
| 5 | \( 1 + (-0.415 - 0.909i)T \) |
| 23 | \( 1 + (-0.755 - 0.654i)T \) |
good | 7 | \( 1 + (-0.540 - 1.84i)T + (-0.841 + 0.540i)T^{2} \) |
| 11 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 13 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 17 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 19 | \( 1 + (-0.959 + 0.281i)T^{2} \) |
| 29 | \( 1 + (1.80 + 0.258i)T + (0.959 + 0.281i)T^{2} \) |
| 31 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 37 | \( 1 + (-0.654 - 0.755i)T^{2} \) |
| 41 | \( 1 + (-0.512 + 0.234i)T + (0.654 - 0.755i)T^{2} \) |
| 43 | \( 1 + (0.153 + 0.239i)T + (-0.415 + 0.909i)T^{2} \) |
| 47 | \( 1 + 0.830iT - T^{2} \) |
| 53 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 59 | \( 1 + (0.841 + 0.540i)T^{2} \) |
| 61 | \( 1 + (-1.07 + 1.66i)T + (-0.415 - 0.909i)T^{2} \) |
| 67 | \( 1 + (-1.27 - 1.10i)T + (0.142 + 0.989i)T^{2} \) |
| 71 | \( 1 + (-0.142 - 0.989i)T^{2} \) |
| 73 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 79 | \( 1 + (0.841 + 0.540i)T^{2} \) |
| 83 | \( 1 + (0.449 - 0.983i)T + (-0.654 - 0.755i)T^{2} \) |
| 89 | \( 1 + (1.10 - 0.708i)T + (0.415 - 0.909i)T^{2} \) |
| 97 | \( 1 + (-0.654 + 0.755i)T^{2} \) |
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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
−9.852186253126202118568692644042, −9.402938107020995965415166218219, −8.459424561418189825701272230894, −7.52153808099766075244721178538, −6.74313103926292616097743583518, −5.77784841391562376143447262640, −5.16950034313904887080162806310, −3.68526751710331210241777300674, −2.45899456830217553058883189435, −1.75521045443057405346142279995,
0.74237363847110342780664175434, 1.63126958592161416582792159436, 4.06423799847423380390137062646, 4.75629937941084783736932863550, 5.48761742826169660546580109384, 6.43473329262893922837838178935, 7.24096023997328368496348002422, 7.80769651370079401254067661714, 8.862995530983414287865990165039, 9.701642364451482885567730669230