L(s) = 1 | − 2.73·2-s + 3-s + 5.47·4-s + 2.29·5-s − 2.73·6-s − 9.49·8-s + 9-s − 6.26·10-s − 0.920·11-s + 5.47·12-s − 6.70·13-s + 2.29·15-s + 15.0·16-s + 3.80·17-s − 2.73·18-s + 6.56·19-s + 12.5·20-s + 2.51·22-s − 6.79·23-s − 9.49·24-s + 0.244·25-s + 18.3·26-s + 27-s − 4.96·29-s − 6.26·30-s + 1.14·31-s − 22.0·32-s + ⋯ |
L(s) = 1 | − 1.93·2-s + 0.577·3-s + 2.73·4-s + 1.02·5-s − 1.11·6-s − 3.35·8-s + 0.333·9-s − 1.97·10-s − 0.277·11-s + 1.58·12-s − 1.86·13-s + 0.591·15-s + 3.75·16-s + 0.923·17-s − 0.644·18-s + 1.50·19-s + 2.80·20-s + 0.536·22-s − 1.41·23-s − 1.93·24-s + 0.0489·25-s + 3.59·26-s + 0.192·27-s − 0.921·29-s − 1.14·30-s + 0.205·31-s − 3.90·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 2 | \( 1 + 2.73T + 2T^{2} \) |
| 5 | \( 1 - 2.29T + 5T^{2} \) |
| 11 | \( 1 + 0.920T + 11T^{2} \) |
| 13 | \( 1 + 6.70T + 13T^{2} \) |
| 17 | \( 1 - 3.80T + 17T^{2} \) |
| 19 | \( 1 - 6.56T + 19T^{2} \) |
| 23 | \( 1 + 6.79T + 23T^{2} \) |
| 29 | \( 1 + 4.96T + 29T^{2} \) |
| 31 | \( 1 - 1.14T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 43 | \( 1 - 3.89T + 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 - 7.81T + 53T^{2} \) |
| 59 | \( 1 - 0.168T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 - 1.48T + 67T^{2} \) |
| 71 | \( 1 + 6.79T + 71T^{2} \) |
| 73 | \( 1 - 4.20T + 73T^{2} \) |
| 79 | \( 1 - 4.67T + 79T^{2} \) |
| 83 | \( 1 + 6.73T + 83T^{2} \) |
| 89 | \( 1 + 8.78T + 89T^{2} \) |
| 97 | \( 1 - 1.94T + 97T^{2} \) |
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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
−7.83258758240338814275732289191, −7.31639360459706657849341836325, −6.75969076925119733245730457778, −5.68055809530208843514586809893, −5.26878065560471884469732828448, −3.58894478372262463224794361459, −2.65881603421315963689920104401, −2.11880884219193505057339618633, −1.34521799157805738551496261810, 0,
1.34521799157805738551496261810, 2.11880884219193505057339618633, 2.65881603421315963689920104401, 3.58894478372262463224794361459, 5.26878065560471884469732828448, 5.68055809530208843514586809893, 6.75969076925119733245730457778, 7.31639360459706657849341836325, 7.83258758240338814275732289191