| L(s) = 1 | − 2.24·2-s + 2.77·3-s + 3.02·4-s + 0.258·5-s − 6.21·6-s + 3.59·7-s − 2.30·8-s + 4.69·9-s − 0.579·10-s + 1.04·11-s + 8.39·12-s + 1.78·13-s − 8.05·14-s + 0.717·15-s − 0.887·16-s + 17-s − 10.5·18-s + 2.63·19-s + 0.783·20-s + 9.96·21-s − 2.34·22-s + 8.20·23-s − 6.39·24-s − 4.93·25-s − 4.01·26-s + 4.69·27-s + 10.8·28-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 1.60·3-s + 1.51·4-s + 0.115·5-s − 2.53·6-s + 1.35·7-s − 0.814·8-s + 1.56·9-s − 0.183·10-s + 0.315·11-s + 2.42·12-s + 0.496·13-s − 2.15·14-s + 0.185·15-s − 0.221·16-s + 0.242·17-s − 2.47·18-s + 0.603·19-s + 0.175·20-s + 2.17·21-s − 0.500·22-s + 1.70·23-s − 1.30·24-s − 0.986·25-s − 0.786·26-s + 0.903·27-s + 2.05·28-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
| \(L(1)\) |
\(\approx\) |
\(2.456479846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.456479846\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.24T + 2T^{2} \) |
| 3 | \( 1 - 2.77T + 3T^{2} \) |
| 5 | \( 1 - 0.258T + 5T^{2} \) |
| 7 | \( 1 - 3.59T + 7T^{2} \) |
| 11 | \( 1 - 1.04T + 11T^{2} \) |
| 13 | \( 1 - 1.78T + 13T^{2} \) |
| 19 | \( 1 - 2.63T + 19T^{2} \) |
| 23 | \( 1 - 8.20T + 23T^{2} \) |
| 29 | \( 1 - 9.95T + 29T^{2} \) |
| 31 | \( 1 - 0.457T + 31T^{2} \) |
| 37 | \( 1 + 0.988T + 37T^{2} \) |
| 41 | \( 1 - 2.03T + 41T^{2} \) |
| 43 | \( 1 - 4.37T + 43T^{2} \) |
| 47 | \( 1 + 7.94T + 47T^{2} \) |
| 53 | \( 1 + 12.8T + 53T^{2} \) |
| 59 | \( 1 - 4.08T + 59T^{2} \) |
| 61 | \( 1 + 6.93T + 61T^{2} \) |
| 67 | \( 1 + 0.0995T + 67T^{2} \) |
| 71 | \( 1 + 2.43T + 71T^{2} \) |
| 73 | \( 1 - 5.94T + 73T^{2} \) |
| 79 | \( 1 - 8.72T + 79T^{2} \) |
| 83 | \( 1 + 8.46T + 83T^{2} \) |
| 89 | \( 1 + 2.36T + 89T^{2} \) |
| 97 | \( 1 + 4.45T + 97T^{2} \) |
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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
−8.074662978456925978814136572546, −7.896275219978740091148052001142, −7.14687290154111265097350731677, −6.38644904192784914376000311005, −5.07785494095723440990005934687, −4.33842739407269858775259062986, −3.25151109753540843979346487073, −2.52532644168084671559810679756, −1.58590165361189228877958950864, −1.12471751715396086280641380986,
1.12471751715396086280641380986, 1.58590165361189228877958950864, 2.52532644168084671559810679756, 3.25151109753540843979346487073, 4.33842739407269858775259062986, 5.07785494095723440990005934687, 6.38644904192784914376000311005, 7.14687290154111265097350731677, 7.896275219978740091148052001142, 8.074662978456925978814136572546
