| L(s) = 1 | − 2.69·2-s + 0.578·3-s + 5.24·4-s + 4.35·5-s − 1.55·6-s − 1.96·7-s − 8.74·8-s − 2.66·9-s − 11.7·10-s + 2.40·11-s + 3.03·12-s + 3.82·13-s + 5.29·14-s + 2.52·15-s + 13.0·16-s + 17-s + 7.17·18-s − 4.69·19-s + 22.8·20-s − 1.13·21-s − 6.48·22-s − 0.242·23-s − 5.06·24-s + 13.9·25-s − 10.3·26-s − 3.27·27-s − 10.3·28-s + ⋯ |
| L(s) = 1 | − 1.90·2-s + 0.334·3-s + 2.62·4-s + 1.94·5-s − 0.636·6-s − 0.743·7-s − 3.09·8-s − 0.888·9-s − 3.70·10-s + 0.725·11-s + 0.877·12-s + 1.06·13-s + 1.41·14-s + 0.651·15-s + 3.26·16-s + 0.242·17-s + 1.69·18-s − 1.07·19-s + 5.11·20-s − 0.248·21-s − 1.38·22-s − 0.0505·23-s − 1.03·24-s + 2.79·25-s − 2.02·26-s − 0.631·27-s − 1.95·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 353 | \( 1 - T \) |
| good | 2 | \( 1 + 2.69T + 2T^{2} \) |
| 3 | \( 1 - 0.578T + 3T^{2} \) |
| 5 | \( 1 - 4.35T + 5T^{2} \) |
| 7 | \( 1 + 1.96T + 7T^{2} \) |
| 11 | \( 1 - 2.40T + 11T^{2} \) |
| 13 | \( 1 - 3.82T + 13T^{2} \) |
| 19 | \( 1 + 4.69T + 19T^{2} \) |
| 23 | \( 1 + 0.242T + 23T^{2} \) |
| 29 | \( 1 - 2.26T + 29T^{2} \) |
| 31 | \( 1 + 9.72T + 31T^{2} \) |
| 37 | \( 1 + 7.73T + 37T^{2} \) |
| 41 | \( 1 + 1.87T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 + 5.16T + 47T^{2} \) |
| 53 | \( 1 + 2.26T + 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 - 1.16T + 61T^{2} \) |
| 67 | \( 1 + 3.92T + 67T^{2} \) |
| 71 | \( 1 + 3.51T + 71T^{2} \) |
| 73 | \( 1 - 16.2T + 73T^{2} \) |
| 79 | \( 1 + 5.97T + 79T^{2} \) |
| 83 | \( 1 + 2.20T + 83T^{2} \) |
| 89 | \( 1 + 6.72T + 89T^{2} \) |
| 97 | \( 1 + 18.8T + 97T^{2} \) |
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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
−8.131058589067991147748787418357, −6.88822635532426788315089124277, −6.49431785410791798193333659372, −6.04198665832182108275712299216, −5.32029656300860183550043202140, −3.46988326341398594749892265126, −2.85664270793435152574684398067, −1.83965362716739832141597908890, −1.50726230818275076869420507756, 0,
1.50726230818275076869420507756, 1.83965362716739832141597908890, 2.85664270793435152574684398067, 3.46988326341398594749892265126, 5.32029656300860183550043202140, 6.04198665832182108275712299216, 6.49431785410791798193333659372, 6.88822635532426788315089124277, 8.131058589067991147748787418357
