L(s) = 1 | − 1.68·5-s + (−0.960 + 2.46i)7-s + 1.24·11-s + (1.96 + 3.39i)13-s + (1.62 + 2.81i)17-s + (−2.36 + 4.09i)19-s − 0.398·23-s − 2.16·25-s + (3.19 − 5.54i)29-s + (−0.289 + 0.500i)31-s + (1.61 − 4.14i)35-s + (2.72 − 4.71i)37-s + (−4.20 − 7.27i)41-s + (−2.46 + 4.26i)43-s + (0.212 + 0.368i)47-s + ⋯ |
L(s) = 1 | − 0.752·5-s + (−0.362 + 0.931i)7-s + 0.375·11-s + (0.543 + 0.941i)13-s + (0.394 + 0.683i)17-s + (−0.541 + 0.938i)19-s − 0.0830·23-s − 0.433·25-s + (0.594 − 1.02i)29-s + (−0.0519 + 0.0899i)31-s + (0.273 − 0.701i)35-s + (0.447 − 0.774i)37-s + (−0.656 − 1.13i)41-s + (−0.375 + 0.650i)43-s + (0.0310 + 0.0537i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.174i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 - 0.174i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6293728482\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6293728482\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.960 - 2.46i)T \) |
good | 5 | \( 1 + 1.68T + 5T^{2} \) |
| 11 | \( 1 - 1.24T + 11T^{2} \) |
| 13 | \( 1 + (-1.96 - 3.39i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.62 - 2.81i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.36 - 4.09i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 0.398T + 23T^{2} \) |
| 29 | \( 1 + (-3.19 + 5.54i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.289 - 0.500i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.72 + 4.71i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.20 + 7.27i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.46 - 4.26i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.212 - 0.368i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.466 - 0.807i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.02 - 5.23i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.10 + 8.84i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.70 - 8.15i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 8.46T + 71T^{2} \) |
| 73 | \( 1 + (-6.82 - 11.8i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.76 + 4.79i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (8.03 - 13.9i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-6.03 + 10.4i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.86 - 10.1i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.993710238421737348907631508254, −8.306158217100995172771725662239, −7.77449288790412925574298143878, −6.64570562234630116980284922372, −6.15572316524848333874594157682, −5.34804815478913862356258663400, −4.07485645804353533198598218333, −3.80057413881095369067822902644, −2.54116297320809723794689597217, −1.52998797709951417165795249310,
0.21163708877284021972454733321, 1.27585291756561469285307149003, 2.91864710666365911582084406015, 3.52750551292125262268147010307, 4.40056322053705449014442824685, 5.14137875520472747105361600539, 6.29335469822051325031570292733, 6.87861626788861099611226284123, 7.67779423282842453187967434207, 8.222031289034292223366198654505