L(s) = 1 | + (2.33 + 1.34i)3-s + (−2.12 + 0.702i)5-s + (−2.90 + 1.67i)7-s + (2.12 + 3.67i)9-s + (−1.62 + 2.81i)11-s + (−1.21 − 3.39i)13-s + (−5.89 − 1.21i)15-s + (−1.68 + 0.974i)17-s + (0.622 + 1.07i)19-s − 9.02·21-s + (−2.33 − 1.34i)23-s + (4.01 − 2.98i)25-s + 3.35i·27-s + (1.5 − 2.59i)29-s − 3.78·31-s + ⋯ |
L(s) = 1 | + (1.34 + 0.777i)3-s + (−0.949 + 0.314i)5-s + (−1.09 + 0.633i)7-s + (0.707 + 1.22i)9-s + (−0.489 + 0.847i)11-s + (−0.337 − 0.941i)13-s + (−1.52 − 0.314i)15-s + (−0.409 + 0.236i)17-s + (0.142 + 0.247i)19-s − 1.96·21-s + (−0.486 − 0.280i)23-s + (0.802 − 0.596i)25-s + 0.645i·27-s + (0.278 − 0.482i)29-s − 0.679·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.127i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.007208820\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.007208820\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (2.12 - 0.702i)T \) |
| 13 | \( 1 + (1.21 + 3.39i)T \) |
good | 3 | \( 1 + (-2.33 - 1.34i)T + (1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (2.90 - 1.67i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.62 - 2.81i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1.68 - 0.974i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.622 - 1.07i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.33 + 1.34i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 3.78T + 31T^{2} \) |
| 37 | \( 1 + (1.68 + 0.974i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.39 - 2.40i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (7.56 - 4.36i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 6.86iT - 47T^{2} \) |
| 53 | \( 1 - 12.8iT - 53T^{2} \) |
| 59 | \( 1 + (-1.26 - 2.19i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.74 - 6.48i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.47 + 2.00i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.62 - 4.54i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 5.46iT - 73T^{2} \) |
| 79 | \( 1 - 13.7T + 79T^{2} \) |
| 83 | \( 1 - 8.61iT - 83T^{2} \) |
| 89 | \( 1 + (-5.15 + 8.93i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (4.56 - 2.63i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.07200844587448307056090514089, −9.555525286539084816290825187424, −8.657118676375645886663625743708, −7.979737959263070909891941719521, −7.27602563889290896839734037459, −6.12661054421170271486569575839, −4.82382332030398054413845785659, −3.92176101526660455933758064540, −3.07140790453103644085971398829, −2.46044794595188180361529456942,
0.35760653168332151520447911181, 2.02359898693882865543271878335, 3.33358027347703448892564524531, 3.67440679048724241384198898490, 5.03964608129877481358664844237, 6.64445239533494609212675733997, 7.06944307686480791996003243885, 7.918325485952659757570139083145, 8.632097730704911204611888292804, 9.239991069137411062457778352505