L(s) = 1 | − i·3-s + (−1 − 2i)5-s − 5i·7-s − 9-s + 5·11-s − i·13-s + (−2 + i)15-s − 3i·17-s + 4·19-s − 5·21-s − 5i·23-s + (−3 + 4i)25-s + i·27-s + 4·29-s − 5i·33-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + (−0.447 − 0.894i)5-s − 1.88i·7-s − 0.333·9-s + 1.50·11-s − 0.277i·13-s + (−0.516 + 0.258i)15-s − 0.727i·17-s + 0.917·19-s − 1.09·21-s − 1.04i·23-s + (−0.600 + 0.800i)25-s + 0.192i·27-s + 0.742·29-s − 0.870i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.614383307\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.614383307\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (1 + 2i)T \) |
| 13 | \( 1 + iT \) |
good | 7 | \( 1 + 5iT - 7T^{2} \) |
| 11 | \( 1 - 5T + 11T^{2} \) |
| 17 | \( 1 + 3iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 5iT - 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 7iT - 37T^{2} \) |
| 41 | \( 1 - 11T + 41T^{2} \) |
| 43 | \( 1 - 12iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 + iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 7T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 7T + 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 - 5T + 79T^{2} \) |
| 83 | \( 1 - 2iT - 83T^{2} \) |
| 89 | \( 1 - 3T + 89T^{2} \) |
| 97 | \( 1 + iT - 97T^{2} \) |
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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
−9.173634983474205144330880674826, −8.091572675578105192101626330397, −7.50242587802434220348209059757, −6.86762031436111040662013913849, −5.97634103260763198120458637157, −4.59839025359759732173204797102, −4.20144596758247429734718963107, −3.11582553573484230142700598824, −1.26926690588574235504681386438, −0.74037234441383962200328963926,
1.82083390214033712196011348908, 2.99063762398255402286193420137, 3.68305852272022394495996522875, 4.77058663860817632306777909911, 5.88390238191610028515611024548, 6.30143934535369195218255744222, 7.37614899805581717834938442773, 8.372128110346176017244995400548, 9.113948906419845761224020086676, 9.552244043282663860021503939118