| L(s) = 1 | + 0.128·3-s + 1.64·5-s − 1.12·7-s − 2.98·9-s + 11-s − 6.41·13-s + 0.211·15-s + 4.45·17-s + 19-s − 0.145·21-s − 3.59·23-s − 2.30·25-s − 0.771·27-s − 1.46·29-s − 7.75·31-s + 0.128·33-s − 1.85·35-s − 1.14·37-s − 0.827·39-s − 5.06·41-s − 0.301·43-s − 4.90·45-s + 8.57·47-s − 5.72·49-s + 0.574·51-s − 9.28·53-s + 1.64·55-s + ⋯ |
| L(s) = 1 | + 0.0744·3-s + 0.734·5-s − 0.426·7-s − 0.994·9-s + 0.301·11-s − 1.77·13-s + 0.0546·15-s + 1.08·17-s + 0.229·19-s − 0.0317·21-s − 0.750·23-s − 0.460·25-s − 0.148·27-s − 0.272·29-s − 1.39·31-s + 0.0224·33-s − 0.313·35-s − 0.188·37-s − 0.132·39-s − 0.791·41-s − 0.0460·43-s − 0.730·45-s + 1.25·47-s − 0.817·49-s + 0.0804·51-s − 1.27·53-s + 0.221·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 0.128T + 3T^{2} \) |
| 5 | \( 1 - 1.64T + 5T^{2} \) |
| 7 | \( 1 + 1.12T + 7T^{2} \) |
| 13 | \( 1 + 6.41T + 13T^{2} \) |
| 17 | \( 1 - 4.45T + 17T^{2} \) |
| 23 | \( 1 + 3.59T + 23T^{2} \) |
| 29 | \( 1 + 1.46T + 29T^{2} \) |
| 31 | \( 1 + 7.75T + 31T^{2} \) |
| 37 | \( 1 + 1.14T + 37T^{2} \) |
| 41 | \( 1 + 5.06T + 41T^{2} \) |
| 43 | \( 1 + 0.301T + 43T^{2} \) |
| 47 | \( 1 - 8.57T + 47T^{2} \) |
| 53 | \( 1 + 9.28T + 53T^{2} \) |
| 59 | \( 1 - 2.85T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 + 0.0460T + 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 - 8.82T + 73T^{2} \) |
| 79 | \( 1 + 9.48T + 79T^{2} \) |
| 83 | \( 1 - 6.97T + 83T^{2} \) |
| 89 | \( 1 - 1.79T + 89T^{2} \) |
| 97 | \( 1 + 7.82T + 97T^{2} \) |
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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
−9.289157367262844285049615339579, −8.088044759867200294608696763143, −7.44261272681243667795418630766, −6.46206992226591274784747442136, −5.62537591393104833408328597070, −5.09134518728388143192534876166, −3.72670884969167580541029397414, −2.79412620124439300736360213500, −1.85458376936106853682358454710, 0,
1.85458376936106853682358454710, 2.79412620124439300736360213500, 3.72670884969167580541029397414, 5.09134518728388143192534876166, 5.62537591393104833408328597070, 6.46206992226591274784747442136, 7.44261272681243667795418630766, 8.088044759867200294608696763143, 9.289157367262844285049615339579
