| L(s) = 1 | − 3.10·3-s − 2.81·5-s + 7-s + 6.62·9-s + 3.10·11-s − 13-s + 8.72·15-s − 0.524·17-s + 0.813·19-s − 3.10·21-s − 7.33·23-s + 2.91·25-s − 11.2·27-s − 8.28·29-s − 1.39·31-s − 9.62·33-s − 2.81·35-s + 6.15·37-s + 3.10·39-s − 4.20·41-s + 6.75·43-s − 18.6·45-s + 5.97·47-s + 49-s + 1.62·51-s + 2.49·53-s − 8.72·55-s + ⋯ |
| L(s) = 1 | − 1.79·3-s − 1.25·5-s + 0.377·7-s + 2.20·9-s + 0.935·11-s − 0.277·13-s + 2.25·15-s − 0.127·17-s + 0.186·19-s − 0.677·21-s − 1.53·23-s + 0.583·25-s − 2.16·27-s − 1.53·29-s − 0.250·31-s − 1.67·33-s − 0.475·35-s + 1.01·37-s + 0.496·39-s − 0.656·41-s + 1.03·43-s − 2.77·45-s + 0.870·47-s + 0.142·49-s + 0.227·51-s + 0.342·53-s − 1.17·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5824 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5824 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 3.10T + 3T^{2} \) |
| 5 | \( 1 + 2.81T + 5T^{2} \) |
| 11 | \( 1 - 3.10T + 11T^{2} \) |
| 17 | \( 1 + 0.524T + 17T^{2} \) |
| 19 | \( 1 - 0.813T + 19T^{2} \) |
| 23 | \( 1 + 7.33T + 23T^{2} \) |
| 29 | \( 1 + 8.28T + 29T^{2} \) |
| 31 | \( 1 + 1.39T + 31T^{2} \) |
| 37 | \( 1 - 6.15T + 37T^{2} \) |
| 41 | \( 1 + 4.20T + 41T^{2} \) |
| 43 | \( 1 - 6.75T + 43T^{2} \) |
| 47 | \( 1 - 5.97T + 47T^{2} \) |
| 53 | \( 1 - 2.49T + 53T^{2} \) |
| 59 | \( 1 + 4.47T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 - 8.72T + 71T^{2} \) |
| 73 | \( 1 + 2.34T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 - 16.4T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 + 1.18T + 97T^{2} \) |
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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
−7.58925538935750885858502324469, −7.01774046625046740800830754677, −6.25377243316185959572864053977, −5.63812007916384627321839317999, −4.88400980258178901774248395358, −4.03414363815320158917673114855, −3.82364097801894604317477357943, −2.05773068506813969950072068484, −0.935715890731264421552198409937, 0,
0.935715890731264421552198409937, 2.05773068506813969950072068484, 3.82364097801894604317477357943, 4.03414363815320158917673114855, 4.88400980258178901774248395358, 5.63812007916384627321839317999, 6.25377243316185959572864053977, 7.01774046625046740800830754677, 7.58925538935750885858502324469
