| L(s) = 1 | + 1.18·3-s + 0.261·5-s − 3.40·7-s − 1.58·9-s − 2.11·11-s + 4.74·13-s + 0.310·15-s + 3.91·17-s + 1.18·19-s − 4.03·21-s + 2.60·23-s − 4.93·25-s − 5.45·27-s + 0.684·29-s − 2.40·31-s − 2.51·33-s − 0.889·35-s − 3.28·37-s + 5.63·39-s + 2.97·41-s + 2.26·43-s − 0.415·45-s − 5.45·47-s + 4.56·49-s + 4.64·51-s − 9.66·53-s − 0.553·55-s + ⋯ |
| L(s) = 1 | + 0.685·3-s + 0.116·5-s − 1.28·7-s − 0.529·9-s − 0.637·11-s + 1.31·13-s + 0.0802·15-s + 0.949·17-s + 0.272·19-s − 0.881·21-s + 0.543·23-s − 0.986·25-s − 1.04·27-s + 0.127·29-s − 0.432·31-s − 0.437·33-s − 0.150·35-s − 0.540·37-s + 0.902·39-s + 0.465·41-s + 0.345·43-s − 0.0619·45-s − 0.795·47-s + 0.652·49-s + 0.651·51-s − 1.32·53-s − 0.0745·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 - 1.18T + 3T^{2} \) |
| 5 | \( 1 - 0.261T + 5T^{2} \) |
| 7 | \( 1 + 3.40T + 7T^{2} \) |
| 11 | \( 1 + 2.11T + 11T^{2} \) |
| 13 | \( 1 - 4.74T + 13T^{2} \) |
| 17 | \( 1 - 3.91T + 17T^{2} \) |
| 19 | \( 1 - 1.18T + 19T^{2} \) |
| 23 | \( 1 - 2.60T + 23T^{2} \) |
| 29 | \( 1 - 0.684T + 29T^{2} \) |
| 31 | \( 1 + 2.40T + 31T^{2} \) |
| 37 | \( 1 + 3.28T + 37T^{2} \) |
| 41 | \( 1 - 2.97T + 41T^{2} \) |
| 43 | \( 1 - 2.26T + 43T^{2} \) |
| 47 | \( 1 + 5.45T + 47T^{2} \) |
| 53 | \( 1 + 9.66T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 - 2.71T + 61T^{2} \) |
| 67 | \( 1 + 3.58T + 67T^{2} \) |
| 71 | \( 1 + 9.71T + 71T^{2} \) |
| 73 | \( 1 + 7.72T + 73T^{2} \) |
| 79 | \( 1 - 1.26T + 79T^{2} \) |
| 83 | \( 1 - 0.519T + 83T^{2} \) |
| 89 | \( 1 - 5.74T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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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
−8.028750363524369750915055129996, −7.57290952491086220232271689755, −6.47735217082443723030356821620, −5.94726445552159095076600806124, −5.25188209513590771225853839866, −3.93254891267759872970246315227, −3.26189669612939567085298202378, −2.79024197409501247673424278457, −1.49780465102691307151329616483, 0,
1.49780465102691307151329616483, 2.79024197409501247673424278457, 3.26189669612939567085298202378, 3.93254891267759872970246315227, 5.25188209513590771225853839866, 5.94726445552159095076600806124, 6.47735217082443723030356821620, 7.57290952491086220232271689755, 8.028750363524369750915055129996
