L(s) = 1 | + (0.623 + 1.07i)2-s + (−0.5 − 0.866i)3-s + (0.222 − 0.385i)4-s + 2.80·5-s + (0.623 − 1.07i)6-s + (−2.40 + 4.15i)7-s + 3.04·8-s + (−0.499 + 0.866i)9-s + (1.74 + 3.02i)10-s + (0.733 + 1.27i)11-s − 0.445·12-s − 5.98·14-s + (−1.40 − 2.42i)15-s + (1.45 + 2.52i)16-s + (1.22 − 2.11i)17-s − 1.24·18-s + ⋯ |
L(s) = 1 | + (0.440 + 0.763i)2-s + (−0.288 − 0.499i)3-s + (0.111 − 0.192i)4-s + 1.25·5-s + (0.254 − 0.440i)6-s + (−0.907 + 1.57i)7-s + 1.07·8-s + (−0.166 + 0.288i)9-s + (0.552 + 0.956i)10-s + (0.221 + 0.383i)11-s − 0.128·12-s − 1.60·14-s + (−0.361 − 0.626i)15-s + (0.363 + 0.630i)16-s + (0.296 − 0.513i)17-s − 0.293·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.668 - 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.668 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.89497 + 0.844313i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.89497 + 0.844313i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.623 - 1.07i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 - 2.80T + 5T^{2} \) |
| 7 | \( 1 + (2.40 - 4.15i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.733 - 1.27i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.22 + 2.11i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.27 + 2.20i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.75 - 3.04i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.925 + 1.60i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 7.63T + 31T^{2} \) |
| 37 | \( 1 + (2.27 + 3.94i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.623 - 1.07i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.19 - 2.06i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 12.8T + 47T^{2} \) |
| 53 | \( 1 + 8.85T + 53T^{2} \) |
| 59 | \( 1 + (-1.08 + 1.88i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.91 + 6.78i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.79 + 3.10i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (4.41 - 7.65i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 7.69T + 73T^{2} \) |
| 79 | \( 1 + 4.02T + 79T^{2} \) |
| 83 | \( 1 - 0.652T + 83T^{2} \) |
| 89 | \( 1 + (-3.14 - 5.45i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.01 - 8.68i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.13829304883064643432139083671, −9.698177270200621749025627711544, −9.587780742091838055243453739645, −8.211035280507374678809399920983, −6.89358483211205139500443500550, −6.33184373849912666968397342099, −5.61839355721030471794321784695, −4.99246656155158216296576287685, −2.83247044062705538197664831192, −1.74987795426988844687777092534,
1.33715567769324276998204945654, 2.99068763832104572656591770281, 3.81405578948723405946821624669, 4.84482290737038705335119133066, 6.17511559297277689867494964616, 6.87428094953980109705763456888, 8.120206645303202718830505546258, 9.527270835483940725275550543456, 10.24477653398774719609639474243, 10.54450073549840625935582253293