| L(s) = 1 | − 2-s + 3-s + 4-s + 3.65·5-s − 6-s + 1.56·7-s − 8-s + 9-s − 3.65·10-s + 0.593·11-s + 12-s + 13-s − 1.56·14-s + 3.65·15-s + 16-s − 3.31·17-s − 18-s − 3.02·19-s + 3.65·20-s + 1.56·21-s − 0.593·22-s + 6.72·23-s − 24-s + 8.33·25-s − 26-s + 27-s + 1.56·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.63·5-s − 0.408·6-s + 0.591·7-s − 0.353·8-s + 0.333·9-s − 1.15·10-s + 0.178·11-s + 0.288·12-s + 0.277·13-s − 0.418·14-s + 0.943·15-s + 0.250·16-s − 0.803·17-s − 0.235·18-s − 0.692·19-s + 0.816·20-s + 0.341·21-s − 0.126·22-s + 1.40·23-s − 0.204·24-s + 1.66·25-s − 0.196·26-s + 0.192·27-s + 0.295·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.125939364\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.125939364\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 5 | \( 1 - 3.65T + 5T^{2} \) |
| 7 | \( 1 - 1.56T + 7T^{2} \) |
| 11 | \( 1 - 0.593T + 11T^{2} \) |
| 17 | \( 1 + 3.31T + 17T^{2} \) |
| 19 | \( 1 + 3.02T + 19T^{2} \) |
| 23 | \( 1 - 6.72T + 23T^{2} \) |
| 29 | \( 1 - 7.76T + 29T^{2} \) |
| 31 | \( 1 - 5.43T + 31T^{2} \) |
| 37 | \( 1 + 2.66T + 37T^{2} \) |
| 41 | \( 1 + 1.56T + 41T^{2} \) |
| 43 | \( 1 - 9.36T + 43T^{2} \) |
| 47 | \( 1 + 9.06T + 47T^{2} \) |
| 53 | \( 1 - 0.780T + 53T^{2} \) |
| 59 | \( 1 + 4.34T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 + 5.60T + 67T^{2} \) |
| 71 | \( 1 + 0.247T + 71T^{2} \) |
| 73 | \( 1 + 13.7T + 73T^{2} \) |
| 79 | \( 1 - 2.31T + 79T^{2} \) |
| 83 | \( 1 + 2.60T + 83T^{2} \) |
| 89 | \( 1 - 9.07T + 89T^{2} \) |
| 97 | \( 1 + 5.85T + 97T^{2} \) |
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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
−8.075752079612606729566678835567, −7.06105103815179311090242304331, −6.54918365046981220276188611081, −5.98411198420436555749942659891, −5.01524311475964436397584368011, −4.44543184547546903127331030080, −3.11321077014795223111576712065, −2.46485749313612027611932158003, −1.75301574390617812804569088490, −1.01662733190656866678937840984,
1.01662733190656866678937840984, 1.75301574390617812804569088490, 2.46485749313612027611932158003, 3.11321077014795223111576712065, 4.44543184547546903127331030080, 5.01524311475964436397584368011, 5.98411198420436555749942659891, 6.54918365046981220276188611081, 7.06105103815179311090242304331, 8.075752079612606729566678835567
