| L(s) = 1 | + 2-s + 3-s + 4-s − 1.54·5-s + 6-s − 2.56·7-s + 8-s + 9-s − 1.54·10-s + 2.03·11-s + 12-s − 1.93·13-s − 2.56·14-s − 1.54·15-s + 16-s + 2.16·17-s + 18-s − 3.78·19-s − 1.54·20-s − 2.56·21-s + 2.03·22-s − 5.52·23-s + 24-s − 2.60·25-s − 1.93·26-s + 27-s − 2.56·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.691·5-s + 0.408·6-s − 0.971·7-s + 0.353·8-s + 0.333·9-s − 0.489·10-s + 0.614·11-s + 0.288·12-s − 0.537·13-s − 0.686·14-s − 0.399·15-s + 0.250·16-s + 0.524·17-s + 0.235·18-s − 0.869·19-s − 0.345·20-s − 0.560·21-s + 0.434·22-s − 1.15·23-s + 0.204·24-s − 0.521·25-s − 0.379·26-s + 0.192·27-s − 0.485·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5766 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5766 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 5 | \( 1 + 1.54T + 5T^{2} \) |
| 7 | \( 1 + 2.56T + 7T^{2} \) |
| 11 | \( 1 - 2.03T + 11T^{2} \) |
| 13 | \( 1 + 1.93T + 13T^{2} \) |
| 17 | \( 1 - 2.16T + 17T^{2} \) |
| 19 | \( 1 + 3.78T + 19T^{2} \) |
| 23 | \( 1 + 5.52T + 23T^{2} \) |
| 29 | \( 1 - 6.88T + 29T^{2} \) |
| 37 | \( 1 - 3.51T + 37T^{2} \) |
| 41 | \( 1 - 7.22T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 - 4.32T + 47T^{2} \) |
| 53 | \( 1 + 2.13T + 53T^{2} \) |
| 59 | \( 1 + 9.10T + 59T^{2} \) |
| 61 | \( 1 - 4.67T + 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 + 5.30T + 71T^{2} \) |
| 73 | \( 1 + 11.9T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 + 18.5T + 89T^{2} \) |
| 97 | \( 1 - 4.45T + 97T^{2} \) |
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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
−7.69867000136222772855242451709, −7.01923948668947049878655653042, −6.30808342537846509958782217642, −5.74096599009706393295126599059, −4.41201826927204599664985344654, −4.19499701909678705783994234340, −3.24427722475753769489659617810, −2.69403394399565253899210643987, −1.55520047455987833931725048777, 0,
1.55520047455987833931725048777, 2.69403394399565253899210643987, 3.24427722475753769489659617810, 4.19499701909678705783994234340, 4.41201826927204599664985344654, 5.74096599009706393295126599059, 6.30808342537846509958782217642, 7.01923948668947049878655653042, 7.69867000136222772855242451709
