| L(s) = 1 | − 2.26·3-s + 3.33·5-s + 2.14·9-s + 4.01·11-s + 5.33·13-s − 7.56·15-s + 17-s − 6.59·19-s − 4.90·23-s + 6.13·25-s + 1.94·27-s − 4.48·29-s − 2.78·31-s − 9.10·33-s + 10.7·37-s − 12.1·39-s + 7.00·41-s + 9.39·43-s + 7.15·45-s + 5.52·47-s − 2.26·51-s − 1.03·53-s + 13.3·55-s + 14.9·57-s + 6.14·59-s + 8.18·61-s + 17.8·65-s + ⋯ |
| L(s) = 1 | − 1.30·3-s + 1.49·5-s + 0.714·9-s + 1.20·11-s + 1.48·13-s − 1.95·15-s + 0.242·17-s − 1.51·19-s − 1.02·23-s + 1.22·25-s + 0.373·27-s − 0.832·29-s − 0.500·31-s − 1.58·33-s + 1.77·37-s − 1.93·39-s + 1.09·41-s + 1.43·43-s + 1.06·45-s + 0.805·47-s − 0.317·51-s − 0.142·53-s + 1.80·55-s + 1.98·57-s + 0.800·59-s + 1.04·61-s + 2.20·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.957545214\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.957545214\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 2.26T + 3T^{2} \) |
| 5 | \( 1 - 3.33T + 5T^{2} \) |
| 11 | \( 1 - 4.01T + 11T^{2} \) |
| 13 | \( 1 - 5.33T + 13T^{2} \) |
| 19 | \( 1 + 6.59T + 19T^{2} \) |
| 23 | \( 1 + 4.90T + 23T^{2} \) |
| 29 | \( 1 + 4.48T + 29T^{2} \) |
| 31 | \( 1 + 2.78T + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 - 7.00T + 41T^{2} \) |
| 43 | \( 1 - 9.39T + 43T^{2} \) |
| 47 | \( 1 - 5.52T + 47T^{2} \) |
| 53 | \( 1 + 1.03T + 53T^{2} \) |
| 59 | \( 1 - 6.14T + 59T^{2} \) |
| 61 | \( 1 - 8.18T + 61T^{2} \) |
| 67 | \( 1 + 9.40T + 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 + 6.36T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 - 7.98T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 - 3.05T + 97T^{2} \) |
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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
−8.002820545629297284504868784218, −6.87440363078634071652632404568, −6.30610101899806120631260929315, −5.81683543417993054472118478421, −5.67460423814397688636626218524, −4.34282747286611515693564847227, −3.93382009501284627277557977741, −2.48862557743352490260883660631, −1.62427244575296290595245472860, −0.824872539977071902808893569630,
0.824872539977071902808893569630, 1.62427244575296290595245472860, 2.48862557743352490260883660631, 3.93382009501284627277557977741, 4.34282747286611515693564847227, 5.67460423814397688636626218524, 5.81683543417993054472118478421, 6.30610101899806120631260929315, 6.87440363078634071652632404568, 8.002820545629297284504868784218
