L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (2.63 + 0.189i)7-s + 0.999·8-s + (2.55 − 1.47i)11-s − 3.93·13-s + (−1.48 + 2.19i)14-s + (−0.5 + 0.866i)16-s + (0.346 − 0.199i)17-s + (−0.0305 − 0.0176i)19-s + 2.94i·22-s + (1.86 − 3.23i)23-s + (1.96 − 3.40i)26-s + (−1.15 − 2.38i)28-s − 8.89i·29-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.997 + 0.0716i)7-s + 0.353·8-s + (0.769 − 0.444i)11-s − 1.09·13-s + (−0.396 + 0.585i)14-s + (−0.125 + 0.216i)16-s + (0.0839 − 0.0484i)17-s + (−0.00700 − 0.00404i)19-s + 0.628i·22-s + (0.389 − 0.673i)23-s + (0.385 − 0.667i)26-s + (−0.218 − 0.449i)28-s − 1.65i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.228i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.590210167\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.590210167\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.63 - 0.189i)T \) |
good | 11 | \( 1 + (-2.55 + 1.47i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 3.93T + 13T^{2} \) |
| 17 | \( 1 + (-0.346 + 0.199i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.0305 + 0.0176i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.86 + 3.23i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 8.89iT - 29T^{2} \) |
| 31 | \( 1 + (-0.717 + 0.414i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.86 - 3.96i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 6.31T + 41T^{2} \) |
| 43 | \( 1 - 3.03iT - 43T^{2} \) |
| 47 | \( 1 + (5.02 + 2.90i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.14 - 3.72i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.78 + 4.82i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (9.97 + 5.76i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (10.8 - 6.25i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 1.93iT - 71T^{2} \) |
| 73 | \( 1 + (-0.171 - 0.297i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.15 + 7.19i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 10.3iT - 83T^{2} \) |
| 89 | \( 1 + (-3.08 + 5.34i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 15.6T + 97T^{2} \) |
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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
−8.550061455194349199153090035443, −7.85764009145863468956871627072, −7.35249866990317926311069411459, −6.35333366379708012101246198196, −5.80226061324493347446889507041, −4.68865964306738015195787952422, −4.37528260554408069981524616218, −2.92375572000627683572015193297, −1.85200370240362113109299328282, −0.64819670111535486085429715790,
1.10186943764415264914703874207, 1.95823042362387622248127077116, 2.94325682858416534763234846787, 4.02407835642068418610903446659, 4.73585042259613754353514265036, 5.44491613572697466052796686311, 6.63960922833112477664604411606, 7.49087711470579911125718243590, 7.85695184872187428494701514578, 9.053810130027626276003055340642