| L(s) = 1 | − 0.141·2-s − 3-s − 1.98·4-s + 0.465·5-s + 0.141·6-s − 1.58·7-s + 0.562·8-s + 9-s − 0.0658·10-s − 4.33·11-s + 1.98·12-s − 3.63·13-s + 0.224·14-s − 0.465·15-s + 3.88·16-s + 17-s − 0.141·18-s − 4.34·19-s − 0.922·20-s + 1.58·21-s + 0.613·22-s − 3.99·23-s − 0.562·24-s − 4.78·25-s + 0.513·26-s − 27-s + 3.14·28-s + ⋯ |
| L(s) = 1 | − 0.0999·2-s − 0.577·3-s − 0.990·4-s + 0.208·5-s + 0.0577·6-s − 0.599·7-s + 0.198·8-s + 0.333·9-s − 0.0208·10-s − 1.30·11-s + 0.571·12-s − 1.00·13-s + 0.0599·14-s − 0.120·15-s + 0.970·16-s + 0.242·17-s − 0.0333·18-s − 0.996·19-s − 0.206·20-s + 0.346·21-s + 0.130·22-s − 0.833·23-s − 0.114·24-s − 0.956·25-s + 0.100·26-s − 0.192·27-s + 0.593·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2877728303\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2877728303\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 + 0.141T + 2T^{2} \) |
| 5 | \( 1 - 0.465T + 5T^{2} \) |
| 7 | \( 1 + 1.58T + 7T^{2} \) |
| 11 | \( 1 + 4.33T + 11T^{2} \) |
| 13 | \( 1 + 3.63T + 13T^{2} \) |
| 19 | \( 1 + 4.34T + 19T^{2} \) |
| 23 | \( 1 + 3.99T + 23T^{2} \) |
| 29 | \( 1 + 1.20T + 29T^{2} \) |
| 31 | \( 1 - 6.70T + 31T^{2} \) |
| 37 | \( 1 + 5.07T + 37T^{2} \) |
| 41 | \( 1 - 4.33T + 41T^{2} \) |
| 43 | \( 1 + 4.52T + 43T^{2} \) |
| 47 | \( 1 + 8.72T + 47T^{2} \) |
| 53 | \( 1 + 6.06T + 53T^{2} \) |
| 59 | \( 1 - 4.77T + 59T^{2} \) |
| 61 | \( 1 - 2.01T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 - 2.29T + 71T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 83 | \( 1 - 9.61T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 + 18.6T + 97T^{2} \) |
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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
−8.197240168687998794323702490313, −7.988452734441663641299257976654, −6.92553742685822882214944145811, −6.12010608004187925766273766226, −5.38118717109456325514264460937, −4.80294707547264392277352338365, −4.01338003135131092358173106261, −2.97934825425630028252717841388, −1.91955401760120525385892413050, −0.30760145404344166404672977373,
0.30760145404344166404672977373, 1.91955401760120525385892413050, 2.97934825425630028252717841388, 4.01338003135131092358173106261, 4.80294707547264392277352338365, 5.38118717109456325514264460937, 6.12010608004187925766273766226, 6.92553742685822882214944145811, 7.988452734441663641299257976654, 8.197240168687998794323702490313
