| L(s) = 1 | − 2.32·2-s − 3-s + 3.39·4-s − 2.33·5-s + 2.32·6-s + 7-s − 3.23·8-s + 9-s + 5.42·10-s − 5.14·11-s − 3.39·12-s + 2.56·13-s − 2.32·14-s + 2.33·15-s + 0.724·16-s − 5.51·17-s − 2.32·18-s − 5.52·19-s − 7.92·20-s − 21-s + 11.9·22-s − 6.43·23-s + 3.23·24-s + 0.452·25-s − 5.95·26-s − 27-s + 3.39·28-s + ⋯ |
| L(s) = 1 | − 1.64·2-s − 0.577·3-s + 1.69·4-s − 1.04·5-s + 0.948·6-s + 0.377·7-s − 1.14·8-s + 0.333·9-s + 1.71·10-s − 1.55·11-s − 0.979·12-s + 0.711·13-s − 0.620·14-s + 0.602·15-s + 0.181·16-s − 1.33·17-s − 0.547·18-s − 1.26·19-s − 1.77·20-s − 0.218·21-s + 2.54·22-s − 1.34·23-s + 0.660·24-s + 0.0905·25-s − 1.16·26-s − 0.192·27-s + 0.641·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.03662558101\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03662558101\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 191 | \( 1 + T \) |
| good | 2 | \( 1 + 2.32T + 2T^{2} \) |
| 5 | \( 1 + 2.33T + 5T^{2} \) |
| 11 | \( 1 + 5.14T + 11T^{2} \) |
| 13 | \( 1 - 2.56T + 13T^{2} \) |
| 17 | \( 1 + 5.51T + 17T^{2} \) |
| 19 | \( 1 + 5.52T + 19T^{2} \) |
| 23 | \( 1 + 6.43T + 23T^{2} \) |
| 29 | \( 1 + 4.01T + 29T^{2} \) |
| 31 | \( 1 + 1.20T + 31T^{2} \) |
| 37 | \( 1 + 0.472T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 - 2.89T + 43T^{2} \) |
| 47 | \( 1 - 7.56T + 47T^{2} \) |
| 53 | \( 1 - 5.54T + 53T^{2} \) |
| 59 | \( 1 - 9.09T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 + 1.76T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 + 6.77T + 79T^{2} \) |
| 83 | \( 1 + 6.33T + 83T^{2} \) |
| 89 | \( 1 - 8.62T + 89T^{2} \) |
| 97 | \( 1 - 13.0T + 97T^{2} \) |
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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
−8.608868506079843790665298991920, −7.74491836332329316559545879018, −7.40246305756458613460647311676, −6.49490544450968430606875637765, −5.72160494837196978184436005788, −4.60134990856253112459630586047, −3.92556386573719372072976899781, −2.49804949651477814155360987460, −1.69579969178645543360902378600, −0.13975388954654411280772615291,
0.13975388954654411280772615291, 1.69579969178645543360902378600, 2.49804949651477814155360987460, 3.92556386573719372072976899781, 4.60134990856253112459630586047, 5.72160494837196978184436005788, 6.49490544450968430606875637765, 7.40246305756458613460647311676, 7.74491836332329316559545879018, 8.608868506079843790665298991920
