L(s) = 1 | − 2.72·2-s + 5.42·4-s + 5-s + 0.930·7-s − 9.34·8-s − 2.72·10-s − 1.45·11-s − 0.247·13-s − 2.53·14-s + 14.6·16-s + 7.85·17-s + 0.303·19-s + 5.42·20-s + 3.96·22-s + 6.91·23-s + 25-s + 0.674·26-s + 5.04·28-s + 4.32·29-s − 5.03·31-s − 21.1·32-s − 21.4·34-s + 0.930·35-s + 8.26·37-s − 0.826·38-s − 9.34·40-s − 0.708·41-s + ⋯ |
L(s) = 1 | − 1.92·2-s + 2.71·4-s + 0.447·5-s + 0.351·7-s − 3.30·8-s − 0.861·10-s − 0.438·11-s − 0.0686·13-s − 0.677·14-s + 3.65·16-s + 1.90·17-s + 0.0695·19-s + 1.21·20-s + 0.845·22-s + 1.44·23-s + 0.200·25-s + 0.132·26-s + 0.954·28-s + 0.803·29-s − 0.903·31-s − 3.73·32-s − 3.67·34-s + 0.157·35-s + 1.35·37-s − 0.134·38-s − 1.47·40-s − 0.110·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9809788626\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9809788626\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 89 | \( 1 - T \) |
good | 2 | \( 1 + 2.72T + 2T^{2} \) |
| 7 | \( 1 - 0.930T + 7T^{2} \) |
| 11 | \( 1 + 1.45T + 11T^{2} \) |
| 13 | \( 1 + 0.247T + 13T^{2} \) |
| 17 | \( 1 - 7.85T + 17T^{2} \) |
| 19 | \( 1 - 0.303T + 19T^{2} \) |
| 23 | \( 1 - 6.91T + 23T^{2} \) |
| 29 | \( 1 - 4.32T + 29T^{2} \) |
| 31 | \( 1 + 5.03T + 31T^{2} \) |
| 37 | \( 1 - 8.26T + 37T^{2} \) |
| 41 | \( 1 + 0.708T + 41T^{2} \) |
| 43 | \( 1 - 4.86T + 43T^{2} \) |
| 47 | \( 1 - 3.15T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 + 1.39T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + 3.64T + 73T^{2} \) |
| 79 | \( 1 + 2.82T + 79T^{2} \) |
| 83 | \( 1 + 3.03T + 83T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
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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
−8.457195829670871025497944595169, −7.78250473359599813106175608956, −7.40672642282549563608533052811, −6.48555802575773660729086545017, −5.77781503495003172131432809896, −4.96245752896934696101687533906, −3.30404164065762934361615387641, −2.64489287223040933805350421894, −1.56132598351586582989754432621, −0.811089549581613235149291939229,
0.811089549581613235149291939229, 1.56132598351586582989754432621, 2.64489287223040933805350421894, 3.30404164065762934361615387641, 4.96245752896934696101687533906, 5.77781503495003172131432809896, 6.48555802575773660729086545017, 7.40672642282549563608533052811, 7.78250473359599813106175608956, 8.457195829670871025497944595169