| L(s) = 1 | − 2.44·3-s + 3.53·5-s − 1.74·7-s + 2.95·9-s + 3.64·11-s + 5.09·13-s − 8.63·15-s + 3.16·19-s + 4.25·21-s − 5.73·23-s + 7.50·25-s + 0.100·27-s + 5.90·29-s − 3.88·31-s − 8.88·33-s − 6.15·35-s + 0.0662·37-s − 12.4·39-s − 10.3·41-s + 2.68·43-s + 10.4·45-s + 11.4·47-s − 3.96·49-s + 3.67·53-s + 12.8·55-s − 7.71·57-s − 4.14·59-s + ⋯ |
| L(s) = 1 | − 1.40·3-s + 1.58·5-s − 0.658·7-s + 0.986·9-s + 1.09·11-s + 1.41·13-s − 2.22·15-s + 0.724·19-s + 0.927·21-s − 1.19·23-s + 1.50·25-s + 0.0193·27-s + 1.09·29-s − 0.697·31-s − 1.54·33-s − 1.04·35-s + 0.0108·37-s − 1.99·39-s − 1.62·41-s + 0.409·43-s + 1.55·45-s + 1.66·47-s − 0.566·49-s + 0.504·53-s + 1.73·55-s − 1.02·57-s − 0.539·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.576135239\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.576135239\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + 2.44T + 3T^{2} \) |
| 5 | \( 1 - 3.53T + 5T^{2} \) |
| 7 | \( 1 + 1.74T + 7T^{2} \) |
| 11 | \( 1 - 3.64T + 11T^{2} \) |
| 13 | \( 1 - 5.09T + 13T^{2} \) |
| 19 | \( 1 - 3.16T + 19T^{2} \) |
| 23 | \( 1 + 5.73T + 23T^{2} \) |
| 29 | \( 1 - 5.90T + 29T^{2} \) |
| 31 | \( 1 + 3.88T + 31T^{2} \) |
| 37 | \( 1 - 0.0662T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 2.68T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 - 3.67T + 53T^{2} \) |
| 59 | \( 1 + 4.14T + 59T^{2} \) |
| 61 | \( 1 + 4.00T + 61T^{2} \) |
| 67 | \( 1 - 8.84T + 67T^{2} \) |
| 71 | \( 1 - 1.48T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 + 2.92T + 83T^{2} \) |
| 89 | \( 1 + 2.32T + 89T^{2} \) |
| 97 | \( 1 + 5.81T + 97T^{2} \) |
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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
−9.209992916576753689106948131292, −8.430433530699955041645893131038, −6.94133216350273718037058987161, −6.43952713107637369773131819478, −5.88860671280895006923881815745, −5.46060279318656367331444123464, −4.31262127979554512139500727857, −3.26821552685109933782760062189, −1.82784941201508168553102361937, −0.930101391205765540118553627686,
0.930101391205765540118553627686, 1.82784941201508168553102361937, 3.26821552685109933782760062189, 4.31262127979554512139500727857, 5.46060279318656367331444123464, 5.88860671280895006923881815745, 6.43952713107637369773131819478, 6.94133216350273718037058987161, 8.430433530699955041645893131038, 9.209992916576753689106948131292
