| L(s) = 1 | − 2.28·2-s − 3.10·3-s + 3.24·4-s + 2.16·5-s + 7.10·6-s − 1.63·7-s − 2.84·8-s + 6.62·9-s − 4.95·10-s − 5.86·11-s − 10.0·12-s + 1.80·13-s + 3.75·14-s − 6.71·15-s + 0.0322·16-s + 17-s − 15.1·18-s − 2.74·19-s + 7.02·20-s + 5.08·21-s + 13.4·22-s − 3.48·23-s + 8.83·24-s − 0.314·25-s − 4.13·26-s − 11.2·27-s − 5.31·28-s + ⋯ |
| L(s) = 1 | − 1.61·2-s − 1.79·3-s + 1.62·4-s + 0.968·5-s + 2.90·6-s − 0.619·7-s − 1.00·8-s + 2.20·9-s − 1.56·10-s − 1.76·11-s − 2.90·12-s + 0.501·13-s + 1.00·14-s − 1.73·15-s + 0.00807·16-s + 0.242·17-s − 3.57·18-s − 0.630·19-s + 1.56·20-s + 1.11·21-s + 2.86·22-s − 0.727·23-s + 1.80·24-s − 0.0628·25-s − 0.811·26-s − 2.16·27-s − 1.00·28-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 | \( 1 + 2.28T + 2T^{2} \) |
| 3 | \( 1 + 3.10T + 3T^{2} \) |
| 5 | \( 1 - 2.16T + 5T^{2} \) |
| 7 | \( 1 + 1.63T + 7T^{2} \) |
| 11 | \( 1 + 5.86T + 11T^{2} \) |
| 13 | \( 1 - 1.80T + 13T^{2} \) |
| 19 | \( 1 + 2.74T + 19T^{2} \) |
| 23 | \( 1 + 3.48T + 23T^{2} \) |
| 29 | \( 1 - 5.15T + 29T^{2} \) |
| 31 | \( 1 - 3.45T + 31T^{2} \) |
| 37 | \( 1 - 5.57T + 37T^{2} \) |
| 41 | \( 1 + 4.42T + 41T^{2} \) |
| 43 | \( 1 - 3.95T + 43T^{2} \) |
| 47 | \( 1 - 1.39T + 47T^{2} \) |
| 53 | \( 1 + 4.72T + 53T^{2} \) |
| 59 | \( 1 - 5.75T + 59T^{2} \) |
| 61 | \( 1 + 5.18T + 61T^{2} \) |
| 67 | \( 1 - 14.6T + 67T^{2} \) |
| 71 | \( 1 - 7.70T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 + 2.92T + 79T^{2} \) |
| 83 | \( 1 + 7.63T + 83T^{2} \) |
| 89 | \( 1 + 8.20T + 89T^{2} \) |
| 97 | \( 1 + 15.6T + 97T^{2} \) |
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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
−7.901968757520065302591322705453, −6.83349683913359997491582937131, −6.46495110563616821381882684823, −5.79217884535078289034949347076, −5.23103073527856129423732443304, −4.29361369897962532651568960263, −2.73879371825229446167600188877, −1.89199092210052928221250775179, −0.851431436350103228371574966645, 0,
0.851431436350103228371574966645, 1.89199092210052928221250775179, 2.73879371825229446167600188877, 4.29361369897962532651568960263, 5.23103073527856129423732443304, 5.79217884535078289034949347076, 6.46495110563616821381882684823, 6.83349683913359997491582937131, 7.901968757520065302591322705453
