L(s) = 1 | − i·3-s + 1.41·5-s + (2.53 + 0.770i)7-s − 9-s − 3.81i·11-s − 6.45i·13-s − 1.41i·15-s − 3.92·17-s − 0.414·19-s + (0.770 − 2.53i)21-s + (−4.75 − 0.614i)23-s − 2.99·25-s + i·27-s + 0.316·29-s − 0.348i·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + 0.633·5-s + (0.956 + 0.291i)7-s − 0.333·9-s − 1.15i·11-s − 1.79i·13-s − 0.365i·15-s − 0.951·17-s − 0.0949·19-s + (0.168 − 0.552i)21-s + (−0.991 − 0.128i)23-s − 0.598·25-s + 0.192i·27-s + 0.0588·29-s − 0.0626i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.411 + 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.411 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.744390774\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.744390774\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 + (-2.53 - 0.770i)T \) |
| 23 | \( 1 + (4.75 + 0.614i)T \) |
good | 5 | \( 1 - 1.41T + 5T^{2} \) |
| 11 | \( 1 + 3.81iT - 11T^{2} \) |
| 13 | \( 1 + 6.45iT - 13T^{2} \) |
| 17 | \( 1 + 3.92T + 17T^{2} \) |
| 19 | \( 1 + 0.414T + 19T^{2} \) |
| 29 | \( 1 - 0.316T + 29T^{2} \) |
| 31 | \( 1 + 0.348iT - 31T^{2} \) |
| 37 | \( 1 + 0.720iT - 37T^{2} \) |
| 41 | \( 1 - 1.82iT - 41T^{2} \) |
| 43 | \( 1 + 8.25iT - 43T^{2} \) |
| 47 | \( 1 - 3.93iT - 47T^{2} \) |
| 53 | \( 1 + 7.37iT - 53T^{2} \) |
| 59 | \( 1 - 5.43iT - 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 + 10.0iT - 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 - 14.7iT - 73T^{2} \) |
| 79 | \( 1 + 4.54iT - 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 - 6.10T + 89T^{2} \) |
| 97 | \( 1 - 6.25T + 97T^{2} \) |
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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
−8.609978558266050003997168825311, −8.285794237987445998417342299687, −7.54300654564910988698547074792, −6.42265179430585950560903958559, −5.70126141567865090395181837584, −5.23160926457717044220874601687, −3.90261084752213954092930836445, −2.74274272111423272551972495658, −1.90639082630990026163346308613, −0.59967432518041880180076092290,
1.73501629108216525056064156837, 2.26304773246169444859256942332, 4.03516340061443986961010894881, 4.42438715021696514024052206249, 5.25208696603709915787479159417, 6.32809669502076410798308461481, 7.01382544355484051685154868292, 7.940712443186669963440020357125, 8.823318668807138507290055477307, 9.525463079529671441297625615194