L(s) = 1 | − 2-s + 4-s + (−2.16 − 0.557i)5-s + (1.39 + 2.24i)7-s − 8-s + (2.16 + 0.557i)10-s + 3.69i·11-s − 0.958·13-s + (−1.39 − 2.24i)14-s + 16-s − 1.53i·17-s − 1.31i·19-s + (−2.16 − 0.557i)20-s − 3.69i·22-s − 7.48·23-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + (−0.968 − 0.249i)5-s + (0.527 + 0.849i)7-s − 0.353·8-s + (0.684 + 0.176i)10-s + 1.11i·11-s − 0.265·13-s + (−0.372 − 0.600i)14-s + 0.250·16-s − 0.372i·17-s − 0.300i·19-s + (−0.484 − 0.124i)20-s − 0.788i·22-s − 1.56·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 + 0.298i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.954 + 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.07040360938\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.07040360938\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.16 + 0.557i)T \) |
| 7 | \( 1 + (-1.39 - 2.24i)T \) |
good | 11 | \( 1 - 3.69iT - 11T^{2} \) |
| 13 | \( 1 + 0.958T + 13T^{2} \) |
| 17 | \( 1 + 1.53iT - 17T^{2} \) |
| 19 | \( 1 + 1.31iT - 19T^{2} \) |
| 23 | \( 1 + 7.48T + 23T^{2} \) |
| 29 | \( 1 - 3.86iT - 29T^{2} \) |
| 31 | \( 1 - 3.04iT - 31T^{2} \) |
| 37 | \( 1 + 3.00iT - 37T^{2} \) |
| 41 | \( 1 - 2.07T + 41T^{2} \) |
| 43 | \( 1 + 4.66iT - 43T^{2} \) |
| 47 | \( 1 + 12.3iT - 47T^{2} \) |
| 53 | \( 1 + 7.82T + 53T^{2} \) |
| 59 | \( 1 - 10.1T + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 1.34iT - 67T^{2} \) |
| 71 | \( 1 + 5.46iT - 71T^{2} \) |
| 73 | \( 1 + 7.94T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 - 0.194iT - 83T^{2} \) |
| 89 | \( 1 + 1.16T + 89T^{2} \) |
| 97 | \( 1 + 9.91T + 97T^{2} \) |
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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
−9.534941622898254491892192567905, −8.752568390476600235899289283181, −8.196662619011263306932753998006, −7.40868838194900721404861257352, −6.83165286931870978209536736823, −5.58254699409707804741699770227, −4.81071097099065662919688949181, −3.87161170770742240587517830551, −2.60521949138963133356459791733, −1.65088804234121396616480553649,
0.03458870286144989768502130417, 1.27204445732567289811523277995, 2.72523704275605439893795528424, 3.79076794101442947219105174347, 4.42997180476143406619832916088, 5.79555607241507744956117900380, 6.55020890691122169635999515578, 7.55674969203260862716872694830, 8.016308664349996361474266434513, 8.491792763224286341637197953545