| L(s) = 1 | − 0.199·3-s − 2.28·5-s + 2.52·7-s − 2.96·9-s − 0.983·11-s − 3.79·13-s + 0.456·15-s + 6.19·17-s + 6.89·19-s − 0.505·21-s + 0.399·23-s + 0.218·25-s + 1.19·27-s − 9.76·29-s + 0.0168·31-s + 0.196·33-s − 5.77·35-s − 4.30·37-s + 0.757·39-s + 0.623·41-s + 7.74·43-s + 6.76·45-s + 5.63·47-s − 0.607·49-s − 1.23·51-s + 8.34·53-s + 2.24·55-s + ⋯ |
| L(s) = 1 | − 0.115·3-s − 1.02·5-s + 0.955·7-s − 0.986·9-s − 0.296·11-s − 1.05·13-s + 0.117·15-s + 1.50·17-s + 1.58·19-s − 0.110·21-s + 0.0832·23-s + 0.0436·25-s + 0.229·27-s − 1.81·29-s + 0.00302·31-s + 0.0341·33-s − 0.976·35-s − 0.707·37-s + 0.121·39-s + 0.0973·41-s + 1.18·43-s + 1.00·45-s + 0.821·47-s − 0.0867·49-s − 0.173·51-s + 1.14·53-s + 0.302·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 113 | \( 1 - T \) |
| good | 3 | \( 1 + 0.199T + 3T^{2} \) |
| 5 | \( 1 + 2.28T + 5T^{2} \) |
| 7 | \( 1 - 2.52T + 7T^{2} \) |
| 11 | \( 1 + 0.983T + 11T^{2} \) |
| 13 | \( 1 + 3.79T + 13T^{2} \) |
| 17 | \( 1 - 6.19T + 17T^{2} \) |
| 19 | \( 1 - 6.89T + 19T^{2} \) |
| 23 | \( 1 - 0.399T + 23T^{2} \) |
| 29 | \( 1 + 9.76T + 29T^{2} \) |
| 31 | \( 1 - 0.0168T + 31T^{2} \) |
| 37 | \( 1 + 4.30T + 37T^{2} \) |
| 41 | \( 1 - 0.623T + 41T^{2} \) |
| 43 | \( 1 - 7.74T + 43T^{2} \) |
| 47 | \( 1 - 5.63T + 47T^{2} \) |
| 53 | \( 1 - 8.34T + 53T^{2} \) |
| 59 | \( 1 + 5.49T + 59T^{2} \) |
| 61 | \( 1 + 5.19T + 61T^{2} \) |
| 67 | \( 1 + 3.13T + 67T^{2} \) |
| 71 | \( 1 + 4.80T + 71T^{2} \) |
| 73 | \( 1 + 0.751T + 73T^{2} \) |
| 79 | \( 1 - 6.61T + 79T^{2} \) |
| 83 | \( 1 - 7.15T + 83T^{2} \) |
| 89 | \( 1 - 1.52T + 89T^{2} \) |
| 97 | \( 1 + 2.93T + 97T^{2} \) |
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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
−7.51528187343570328364173022710, −7.40224239091665268066643489522, −5.93458489345699501674467634999, −5.35542176591808009513851264258, −4.91509837638605572225262638823, −3.85439328890103063392669859421, −3.24169847257898307016625904910, −2.34915472899933041981734819419, −1.14460979408063307613595389355, 0,
1.14460979408063307613595389355, 2.34915472899933041981734819419, 3.24169847257898307016625904910, 3.85439328890103063392669859421, 4.91509837638605572225262638823, 5.35542176591808009513851264258, 5.93458489345699501674467634999, 7.40224239091665268066643489522, 7.51528187343570328364173022710
