| L(s) = 1 | − 1.59·2-s − 3-s + 0.550·4-s + 0.890·5-s + 1.59·6-s − 7-s + 2.31·8-s + 9-s − 1.42·10-s + 5.65·11-s − 0.550·12-s − 5.71·13-s + 1.59·14-s − 0.890·15-s − 4.79·16-s − 5.71·17-s − 1.59·18-s − 4.02·19-s + 0.489·20-s + 21-s − 9.02·22-s + 6.76·23-s − 2.31·24-s − 4.20·25-s + 9.13·26-s − 27-s − 0.550·28-s + ⋯ |
| L(s) = 1 | − 1.12·2-s − 0.577·3-s + 0.275·4-s + 0.398·5-s + 0.651·6-s − 0.377·7-s + 0.818·8-s + 0.333·9-s − 0.449·10-s + 1.70·11-s − 0.158·12-s − 1.58·13-s + 0.426·14-s − 0.229·15-s − 1.19·16-s − 1.38·17-s − 0.376·18-s − 0.922·19-s + 0.109·20-s + 0.218·21-s − 1.92·22-s + 1.41·23-s − 0.472·24-s − 0.841·25-s + 1.79·26-s − 0.192·27-s − 0.103·28-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
| \(L(1)\) |
\(\approx\) |
\(0.6004151484\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6004151484\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.59T + 2T^{2} \) |
| 5 | \( 1 - 0.890T + 5T^{2} \) |
| 11 | \( 1 - 5.65T + 11T^{2} \) |
| 13 | \( 1 + 5.71T + 13T^{2} \) |
| 17 | \( 1 + 5.71T + 17T^{2} \) |
| 19 | \( 1 + 4.02T + 19T^{2} \) |
| 23 | \( 1 - 6.76T + 23T^{2} \) |
| 29 | \( 1 + 5.32T + 29T^{2} \) |
| 31 | \( 1 - 7.77T + 31T^{2} \) |
| 37 | \( 1 + 8.51T + 37T^{2} \) |
| 41 | \( 1 - 8.78T + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 + 1.23T + 53T^{2} \) |
| 59 | \( 1 + 7.61T + 59T^{2} \) |
| 61 | \( 1 + 2.73T + 61T^{2} \) |
| 67 | \( 1 - 5.16T + 67T^{2} \) |
| 71 | \( 1 + 5.29T + 71T^{2} \) |
| 73 | \( 1 + 9.05T + 73T^{2} \) |
| 79 | \( 1 + 8.36T + 79T^{2} \) |
| 83 | \( 1 - 0.406T + 83T^{2} \) |
| 89 | \( 1 - 0.301T + 89T^{2} \) |
| 97 | \( 1 + 5.52T + 97T^{2} \) |
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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
−8.913298045087153683395782437707, −7.61916990826120378045370868066, −7.03940504939318419943392920303, −6.50596987388476408530497695844, −5.63929900278929152327940012978, −4.47477862780677958540188442575, −4.19904527377662197740527326306, −2.56322995002330056975449093158, −1.68960831142684343486949010306, −0.55074900238463661709639062572,
0.55074900238463661709639062572, 1.68960831142684343486949010306, 2.56322995002330056975449093158, 4.19904527377662197740527326306, 4.47477862780677958540188442575, 5.63929900278929152327940012978, 6.50596987388476408530497695844, 7.03940504939318419943392920303, 7.61916990826120378045370868066, 8.913298045087153683395782437707
