| L(s) = 1 | + 1.06·2-s + 3.07·3-s − 0.863·4-s − 5-s + 3.27·6-s − 3.05·8-s + 6.45·9-s − 1.06·10-s + 11-s − 2.65·12-s + 2.06·13-s − 3.07·15-s − 1.52·16-s − 1.12·17-s + 6.88·18-s + 8.47·19-s + 0.863·20-s + 1.06·22-s + 3.99·23-s − 9.38·24-s + 25-s + 2.19·26-s + 10.6·27-s − 4.01·29-s − 3.27·30-s − 0.325·31-s + 4.47·32-s + ⋯ |
| L(s) = 1 | + 0.753·2-s + 1.77·3-s − 0.431·4-s − 0.447·5-s + 1.33·6-s − 1.07·8-s + 2.15·9-s − 0.337·10-s + 0.301·11-s − 0.766·12-s + 0.571·13-s − 0.793·15-s − 0.381·16-s − 0.272·17-s + 1.62·18-s + 1.94·19-s + 0.193·20-s + 0.227·22-s + 0.833·23-s − 1.91·24-s + 0.200·25-s + 0.431·26-s + 2.04·27-s − 0.746·29-s − 0.598·30-s − 0.0584·31-s + 0.791·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.316699949\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.316699949\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 1.06T + 2T^{2} \) |
| 3 | \( 1 - 3.07T + 3T^{2} \) |
| 13 | \( 1 - 2.06T + 13T^{2} \) |
| 17 | \( 1 + 1.12T + 17T^{2} \) |
| 19 | \( 1 - 8.47T + 19T^{2} \) |
| 23 | \( 1 - 3.99T + 23T^{2} \) |
| 29 | \( 1 + 4.01T + 29T^{2} \) |
| 31 | \( 1 + 0.325T + 31T^{2} \) |
| 37 | \( 1 - 5.40T + 37T^{2} \) |
| 41 | \( 1 - 1.23T + 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 + 1.05T + 47T^{2} \) |
| 53 | \( 1 + 0.884T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 - 1.63T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 - 8.18T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 - 7.43T + 79T^{2} \) |
| 83 | \( 1 - 4.20T + 83T^{2} \) |
| 89 | \( 1 + 14.0T + 89T^{2} \) |
| 97 | \( 1 + 16.8T + 97T^{2} \) |
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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
−8.699557308159523010896205301450, −8.279388561283877232662030235120, −7.43140818348485066430873653223, −6.72843255095321179331884300140, −5.49356595315056355889287700208, −4.68444760939162199450965612115, −3.71979819502413525724154825599, −3.43277524322237454216086910803, −2.55938562016275342327346314263, −1.17389673569543219227284722732,
1.17389673569543219227284722732, 2.55938562016275342327346314263, 3.43277524322237454216086910803, 3.71979819502413525724154825599, 4.68444760939162199450965612115, 5.49356595315056355889287700208, 6.72843255095321179331884300140, 7.43140818348485066430873653223, 8.279388561283877232662030235120, 8.699557308159523010896205301450
