| L(s) = 1 | + 2-s + 3-s + 4-s + 3.42·5-s + 6-s + 2.22·7-s + 8-s + 9-s + 3.42·10-s − 11-s + 12-s − 0.881·13-s + 2.22·14-s + 3.42·15-s + 16-s − 3.62·17-s + 18-s + 4.42·19-s + 3.42·20-s + 2.22·21-s − 22-s − 4.04·23-s + 24-s + 6.74·25-s − 0.881·26-s + 27-s + 2.22·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.53·5-s + 0.408·6-s + 0.840·7-s + 0.353·8-s + 0.333·9-s + 1.08·10-s − 0.301·11-s + 0.288·12-s − 0.244·13-s + 0.594·14-s + 0.884·15-s + 0.250·16-s − 0.879·17-s + 0.235·18-s + 1.01·19-s + 0.766·20-s + 0.485·21-s − 0.213·22-s − 0.843·23-s + 0.204·24-s + 1.34·25-s − 0.172·26-s + 0.192·27-s + 0.420·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.567599631\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.567599631\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 5 | \( 1 - 3.42T + 5T^{2} \) |
| 7 | \( 1 - 2.22T + 7T^{2} \) |
| 13 | \( 1 + 0.881T + 13T^{2} \) |
| 17 | \( 1 + 3.62T + 17T^{2} \) |
| 19 | \( 1 - 4.42T + 19T^{2} \) |
| 23 | \( 1 + 4.04T + 23T^{2} \) |
| 29 | \( 1 - 3.86T + 29T^{2} \) |
| 31 | \( 1 - 2.61T + 31T^{2} \) |
| 37 | \( 1 + 5.62T + 37T^{2} \) |
| 41 | \( 1 + 6.07T + 41T^{2} \) |
| 43 | \( 1 - 2.54T + 43T^{2} \) |
| 47 | \( 1 - 13.3T + 47T^{2} \) |
| 53 | \( 1 + 6.63T + 53T^{2} \) |
| 59 | \( 1 + 1.91T + 59T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 + 14.4T + 83T^{2} \) |
| 89 | \( 1 + 0.582T + 89T^{2} \) |
| 97 | \( 1 - 3.96T + 97T^{2} \) |
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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
−8.424810469799352886900560600818, −7.69031028614462494222097955814, −6.83548600091655921971680222151, −6.16327752957293997297298008985, −5.29705176619497857720930283156, −4.87334672684368575733860459109, −3.87727918483548388179403882251, −2.74795188337003411605783503529, −2.16600607829489477691530340275, −1.36273614547147086299217959479,
1.36273614547147086299217959479, 2.16600607829489477691530340275, 2.74795188337003411605783503529, 3.87727918483548388179403882251, 4.87334672684368575733860459109, 5.29705176619497857720930283156, 6.16327752957293997297298008985, 6.83548600091655921971680222151, 7.69031028614462494222097955814, 8.424810469799352886900560600818
