| L(s) = 1 | − 1.56·2-s − 3-s + 0.459·4-s + 2.88·5-s + 1.56·6-s + 7-s + 2.41·8-s + 9-s − 4.52·10-s + 0.607·11-s − 0.459·12-s + 2.94·13-s − 1.56·14-s − 2.88·15-s − 4.70·16-s − 1.56·18-s − 4.36·19-s + 1.32·20-s − 21-s − 0.953·22-s + 1.57·23-s − 2.41·24-s + 3.33·25-s − 4.61·26-s − 27-s + 0.459·28-s − 6.71·29-s + ⋯ |
| L(s) = 1 | − 1.10·2-s − 0.577·3-s + 0.229·4-s + 1.29·5-s + 0.640·6-s + 0.377·7-s + 0.854·8-s + 0.333·9-s − 1.43·10-s + 0.183·11-s − 0.132·12-s + 0.816·13-s − 0.419·14-s − 0.745·15-s − 1.17·16-s − 0.369·18-s − 1.00·19-s + 0.296·20-s − 0.218·21-s − 0.203·22-s + 0.328·23-s − 0.493·24-s + 0.666·25-s − 0.905·26-s − 0.192·27-s + 0.0869·28-s − 1.24·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6069 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6069 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.160024413\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.160024413\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 1.56T + 2T^{2} \) |
| 5 | \( 1 - 2.88T + 5T^{2} \) |
| 11 | \( 1 - 0.607T + 11T^{2} \) |
| 13 | \( 1 - 2.94T + 13T^{2} \) |
| 19 | \( 1 + 4.36T + 19T^{2} \) |
| 23 | \( 1 - 1.57T + 23T^{2} \) |
| 29 | \( 1 + 6.71T + 29T^{2} \) |
| 31 | \( 1 - 7.18T + 31T^{2} \) |
| 37 | \( 1 + 3.24T + 37T^{2} \) |
| 41 | \( 1 + 3.67T + 41T^{2} \) |
| 43 | \( 1 - 3.91T + 43T^{2} \) |
| 47 | \( 1 - 5.16T + 47T^{2} \) |
| 53 | \( 1 + 1.08T + 53T^{2} \) |
| 59 | \( 1 + 5.98T + 59T^{2} \) |
| 61 | \( 1 + 9.96T + 61T^{2} \) |
| 67 | \( 1 - 6.64T + 67T^{2} \) |
| 71 | \( 1 - 7.11T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 - 7.08T + 79T^{2} \) |
| 83 | \( 1 - 3.31T + 83T^{2} \) |
| 89 | \( 1 - 8.33T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−8.180889392508410810711306904578, −7.50951216163079842634825218197, −6.54924352918586400720863552986, −6.16130449645785359267718680337, −5.28069936214884055958224171634, −4.62738657883719211306489304969, −3.69025868342638801496566141616, −2.23860932211185329358701043707, −1.63988977204608327120081058345, −0.73034723404557306377468884047,
0.73034723404557306377468884047, 1.63988977204608327120081058345, 2.23860932211185329358701043707, 3.69025868342638801496566141616, 4.62738657883719211306489304969, 5.28069936214884055958224171634, 6.16130449645785359267718680337, 6.54924352918586400720863552986, 7.50951216163079842634825218197, 8.180889392508410810711306904578
