| L(s) = 1 | − 1.61·2-s + 3-s + 0.618·4-s − 1.61·6-s − 3·7-s + 2.23·8-s + 9-s + 0.618·12-s + 1.85·13-s + 4.85·14-s − 4.85·16-s − 7.47·17-s − 1.61·18-s + 2.76·19-s − 3·21-s + 7.61·23-s + 2.23·24-s − 3·26-s + 27-s − 1.85·28-s − 3.61·29-s − 8.85·31-s + 3.38·32-s + 12.0·34-s + 0.618·36-s + 8.85·37-s − 4.47·38-s + ⋯ |
| L(s) = 1 | − 1.14·2-s + 0.577·3-s + 0.309·4-s − 0.660·6-s − 1.13·7-s + 0.790·8-s + 0.333·9-s + 0.178·12-s + 0.514·13-s + 1.29·14-s − 1.21·16-s − 1.81·17-s − 0.381·18-s + 0.634·19-s − 0.654·21-s + 1.58·23-s + 0.456·24-s − 0.588·26-s + 0.192·27-s − 0.350·28-s − 0.671·29-s − 1.59·31-s + 0.597·32-s + 2.07·34-s + 0.103·36-s + 1.45·37-s − 0.725·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 13 | \( 1 - 1.85T + 13T^{2} \) |
| 17 | \( 1 + 7.47T + 17T^{2} \) |
| 19 | \( 1 - 2.76T + 19T^{2} \) |
| 23 | \( 1 - 7.61T + 23T^{2} \) |
| 29 | \( 1 + 3.61T + 29T^{2} \) |
| 31 | \( 1 + 8.85T + 31T^{2} \) |
| 37 | \( 1 - 8.85T + 37T^{2} \) |
| 41 | \( 1 - 0.236T + 41T^{2} \) |
| 43 | \( 1 + 1.76T + 43T^{2} \) |
| 47 | \( 1 - 9.38T + 47T^{2} \) |
| 53 | \( 1 + 0.472T + 53T^{2} \) |
| 59 | \( 1 + 4.14T + 59T^{2} \) |
| 61 | \( 1 - 5.76T + 61T^{2} \) |
| 67 | \( 1 + 8.70T + 67T^{2} \) |
| 71 | \( 1 - 1.47T + 71T^{2} \) |
| 73 | \( 1 - 6.85T + 73T^{2} \) |
| 79 | \( 1 - 7.56T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 - 2.56T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−7.39401641156667914957717249581, −7.00825528403734035525457473978, −6.33178366059962705640267396890, −5.36913166120561568995693164280, −4.43148619366494107008439830845, −3.75688340154129936556983250799, −2.89388583190641957976203503624, −2.08335802483974121077697791137, −1.04902840596070642649305272918, 0,
1.04902840596070642649305272918, 2.08335802483974121077697791137, 2.89388583190641957976203503624, 3.75688340154129936556983250799, 4.43148619366494107008439830845, 5.36913166120561568995693164280, 6.33178366059962705640267396890, 7.00825528403734035525457473978, 7.39401641156667914957717249581
