| L(s) = 1 | − 2-s − 0.909·3-s + 4-s + 0.734·5-s + 0.909·6-s + 5.27·7-s − 8-s − 2.17·9-s − 0.734·10-s + 2.39·11-s − 0.909·12-s + 3.37·13-s − 5.27·14-s − 0.668·15-s + 16-s − 0.0921·17-s + 2.17·18-s − 0.769·19-s + 0.734·20-s − 4.79·21-s − 2.39·22-s + 23-s + 0.909·24-s − 4.45·25-s − 3.37·26-s + 4.70·27-s + 5.27·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.525·3-s + 0.5·4-s + 0.328·5-s + 0.371·6-s + 1.99·7-s − 0.353·8-s − 0.724·9-s − 0.232·10-s + 0.723·11-s − 0.262·12-s + 0.937·13-s − 1.40·14-s − 0.172·15-s + 0.250·16-s − 0.0223·17-s + 0.512·18-s − 0.176·19-s + 0.164·20-s − 1.04·21-s − 0.511·22-s + 0.208·23-s + 0.185·24-s − 0.891·25-s − 0.662·26-s + 0.905·27-s + 0.996·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.760204515\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.760204515\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 + 0.909T + 3T^{2} \) |
| 5 | \( 1 - 0.734T + 5T^{2} \) |
| 7 | \( 1 - 5.27T + 7T^{2} \) |
| 11 | \( 1 - 2.39T + 11T^{2} \) |
| 13 | \( 1 - 3.37T + 13T^{2} \) |
| 17 | \( 1 + 0.0921T + 17T^{2} \) |
| 19 | \( 1 + 0.769T + 19T^{2} \) |
| 29 | \( 1 + 1.18T + 29T^{2} \) |
| 31 | \( 1 - 6.49T + 31T^{2} \) |
| 37 | \( 1 - 7.95T + 37T^{2} \) |
| 41 | \( 1 + 1.03T + 41T^{2} \) |
| 43 | \( 1 + 1.18T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 - 6.70T + 53T^{2} \) |
| 59 | \( 1 + 6.08T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 - 7.09T + 67T^{2} \) |
| 71 | \( 1 + 5.87T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 - 4.09T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 - 2.83T + 97T^{2} \) |
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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
−8.068537045868498689719563518444, −7.69650388627627473175957618937, −6.48247179681849156960589184765, −6.11887418214012567092217965942, −5.25331780608389312023732043722, −4.62996129643093747604952894599, −3.67253551583660753885428006585, −2.43770047561563982451851713098, −1.60949460388970029789329942252, −0.866836839441114412059039342585,
0.866836839441114412059039342585, 1.60949460388970029789329942252, 2.43770047561563982451851713098, 3.67253551583660753885428006585, 4.62996129643093747604952894599, 5.25331780608389312023732043722, 6.11887418214012567092217965942, 6.48247179681849156960589184765, 7.69650388627627473175957618937, 8.068537045868498689719563518444
