| L(s) = 1 | + 1.82·3-s + 2.84·5-s + 3.25·7-s + 0.336·9-s − 0.241·11-s − 13-s + 5.19·15-s + 5.04·17-s − 1.20·19-s + 5.93·21-s + 7.89·23-s + 3.08·25-s − 4.86·27-s + 29-s + 1.85·31-s − 0.441·33-s + 9.24·35-s + 0.584·37-s − 1.82·39-s − 4.04·41-s + 1.75·43-s + 0.957·45-s − 7.07·47-s + 3.56·49-s + 9.21·51-s + 7.29·53-s − 0.686·55-s + ⋯ |
| L(s) = 1 | + 1.05·3-s + 1.27·5-s + 1.22·7-s + 0.112·9-s − 0.0728·11-s − 0.277·13-s + 1.34·15-s + 1.22·17-s − 0.277·19-s + 1.29·21-s + 1.64·23-s + 0.617·25-s − 0.936·27-s + 0.185·29-s + 0.333·31-s − 0.0767·33-s + 1.56·35-s + 0.0961·37-s − 0.292·39-s − 0.632·41-s + 0.268·43-s + 0.142·45-s − 1.03·47-s + 0.509·49-s + 1.29·51-s + 1.00·53-s − 0.0926·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.626439036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.626439036\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 1.82T + 3T^{2} \) |
| 5 | \( 1 - 2.84T + 5T^{2} \) |
| 7 | \( 1 - 3.25T + 7T^{2} \) |
| 11 | \( 1 + 0.241T + 11T^{2} \) |
| 17 | \( 1 - 5.04T + 17T^{2} \) |
| 19 | \( 1 + 1.20T + 19T^{2} \) |
| 23 | \( 1 - 7.89T + 23T^{2} \) |
| 31 | \( 1 - 1.85T + 31T^{2} \) |
| 37 | \( 1 - 0.584T + 37T^{2} \) |
| 41 | \( 1 + 4.04T + 41T^{2} \) |
| 43 | \( 1 - 1.75T + 43T^{2} \) |
| 47 | \( 1 + 7.07T + 47T^{2} \) |
| 53 | \( 1 - 7.29T + 53T^{2} \) |
| 59 | \( 1 + 7.06T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 + 4.31T + 67T^{2} \) |
| 71 | \( 1 + 3.00T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 + 8.80T + 79T^{2} \) |
| 83 | \( 1 - 4.97T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 - 18.3T + 97T^{2} \) |
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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
−8.100542731907831559970757918098, −7.58719893757801000117201967101, −6.74904206351610975594457665473, −5.78017504013554801864758393279, −5.24212397614342263469547040331, −4.55074648255295447633818391304, −3.38475772577054054207551381529, −2.69666015059788684392959126815, −1.92851167282776198195295058468, −1.19411563934606120978688305648,
1.19411563934606120978688305648, 1.92851167282776198195295058468, 2.69666015059788684392959126815, 3.38475772577054054207551381529, 4.55074648255295447633818391304, 5.24212397614342263469547040331, 5.78017504013554801864758393279, 6.74904206351610975594457665473, 7.58719893757801000117201967101, 8.100542731907831559970757918098
