| L(s) = 1 | − 1.61·2-s + 0.618·4-s + 2.61·5-s + 3.85·7-s + 2.23·8-s − 4.23·10-s + 3·11-s − 5.47·13-s − 6.23·14-s − 4.85·16-s − 5.23·17-s − 5.85·19-s + 1.61·20-s − 4.85·22-s + 4.09·23-s + 1.85·25-s + 8.85·26-s + 2.38·28-s − 3·31-s + 3.38·32-s + 8.47·34-s + 10.0·35-s − 3.70·37-s + 9.47·38-s + 5.85·40-s + 5.76·41-s + 4.85·43-s + ⋯ |
| L(s) = 1 | − 1.14·2-s + 0.309·4-s + 1.17·5-s + 1.45·7-s + 0.790·8-s − 1.33·10-s + 0.904·11-s − 1.51·13-s − 1.66·14-s − 1.21·16-s − 1.26·17-s − 1.34·19-s + 0.361·20-s − 1.03·22-s + 0.852·23-s + 0.370·25-s + 1.73·26-s + 0.450·28-s − 0.538·31-s + 0.597·32-s + 1.45·34-s + 1.70·35-s − 0.609·37-s + 1.53·38-s + 0.925·40-s + 0.900·41-s + 0.740·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 5 | \( 1 - 2.61T + 5T^{2} \) |
| 7 | \( 1 - 3.85T + 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + 5.47T + 13T^{2} \) |
| 17 | \( 1 + 5.23T + 17T^{2} \) |
| 19 | \( 1 + 5.85T + 19T^{2} \) |
| 23 | \( 1 - 4.09T + 23T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 + 3.70T + 37T^{2} \) |
| 41 | \( 1 - 5.76T + 41T^{2} \) |
| 43 | \( 1 - 4.85T + 43T^{2} \) |
| 47 | \( 1 + 9.38T + 47T^{2} \) |
| 53 | \( 1 - 1.52T + 53T^{2} \) |
| 59 | \( 1 - 1.90T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 + 7T + 67T^{2} \) |
| 71 | \( 1 + 1.14T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 + 7.76T + 79T^{2} \) |
| 83 | \( 1 + 7.94T + 83T^{2} \) |
| 89 | \( 1 + 13.9T + 89T^{2} \) |
| 97 | \( 1 + 3.70T + 97T^{2} \) |
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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
−7.56411314941463728486485319268, −7.08187971261500196300632401788, −6.32335747786494892867448021083, −5.40997908342432592544320600810, −4.57860311610881536936892875855, −4.34736707620158582432674652417, −2.59066596421839611463931406048, −1.89416219638705577718539225898, −1.42987578127935682064971696443, 0,
1.42987578127935682064971696443, 1.89416219638705577718539225898, 2.59066596421839611463931406048, 4.34736707620158582432674652417, 4.57860311610881536936892875855, 5.40997908342432592544320600810, 6.32335747786494892867448021083, 7.08187971261500196300632401788, 7.56411314941463728486485319268
