L(s) = 1 | + 1.57i·3-s − 49i·7-s + 240.·9-s + 322.·11-s + 657. i·13-s − 665. i·17-s + 2.21e3·19-s + 77.3·21-s + 1.58e3i·23-s + 763. i·27-s − 6.38e3·29-s + 7.71e3·31-s + 509. i·33-s − 3.43e3i·37-s − 1.03e3·39-s + ⋯ |
L(s) = 1 | + 0.101i·3-s − 0.377i·7-s + 0.989·9-s + 0.804·11-s + 1.07i·13-s − 0.558i·17-s + 1.40·19-s + 0.0382·21-s + 0.625i·23-s + 0.201i·27-s − 1.41·29-s + 1.44·31-s + 0.0814i·33-s − 0.412i·37-s − 0.109·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.724093502\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.724093502\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 49iT \) |
good | 3 | \( 1 - 1.57iT - 243T^{2} \) |
| 11 | \( 1 - 322.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 657. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 665. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 2.21e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.58e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 6.38e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.71e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.43e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 6.53e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.66e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.25e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 4.24e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 2.61e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.26e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.15e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 3.38e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.21e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 3.52e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.75e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 7.28e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.66e5iT - 8.58e9T^{2} \) |
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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
−9.499340298393855367704736020803, −9.291439422497148848854007282137, −7.82857541631037873905344433437, −7.13474328882530128501090294950, −6.37911322901830627403363950449, −5.09245439138985207923524462621, −4.21640115367422201315069247247, −3.35915323752538163270186320249, −1.81570999956989199154071814285, −0.935261937219353378877836082902,
0.72911885323636821140721839518, 1.71484268156701881966570971454, 3.05798328581133508057363590606, 4.04472661761417354231616865350, 5.14171379280802124512533229010, 6.07464537048818814302180258331, 7.03271910214364520705850199251, 7.84933666250167715147436787004, 8.758352385043294761698034474623, 9.736142875789727026441194762662