| L(s) = 1 | − 0.381·2-s − 3-s − 1.85·4-s + 1.61·5-s + 0.381·6-s − 7-s + 1.47·8-s + 9-s − 0.618·10-s + 1.85·12-s − 4.23·13-s + 0.381·14-s − 1.61·15-s + 3.14·16-s − 7.85·17-s − 0.381·18-s − 0.854·19-s − 3·20-s + 21-s − 4.23·23-s − 1.47·24-s − 2.38·25-s + 1.61·26-s − 27-s + 1.85·28-s − 6·29-s + 0.618·30-s + ⋯ |
| L(s) = 1 | − 0.270·2-s − 0.577·3-s − 0.927·4-s + 0.723·5-s + 0.155·6-s − 0.377·7-s + 0.520·8-s + 0.333·9-s − 0.195·10-s + 0.535·12-s − 1.17·13-s + 0.102·14-s − 0.417·15-s + 0.786·16-s − 1.90·17-s − 0.0900·18-s − 0.195·19-s − 0.670·20-s + 0.218·21-s − 0.883·23-s − 0.300·24-s − 0.476·25-s + 0.317·26-s − 0.192·27-s + 0.350·28-s − 1.11·29-s + 0.112·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.381T + 2T^{2} \) |
| 5 | \( 1 - 1.61T + 5T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 13 | \( 1 + 4.23T + 13T^{2} \) |
| 17 | \( 1 + 7.85T + 17T^{2} \) |
| 19 | \( 1 + 0.854T + 19T^{2} \) |
| 23 | \( 1 + 4.23T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 5.09T + 31T^{2} \) |
| 37 | \( 1 + 1.76T + 37T^{2} \) |
| 41 | \( 1 + 4.23T + 41T^{2} \) |
| 43 | \( 1 - 6.70T + 43T^{2} \) |
| 47 | \( 1 - 1.09T + 47T^{2} \) |
| 53 | \( 1 + 2.61T + 53T^{2} \) |
| 59 | \( 1 - 9.61T + 59T^{2} \) |
| 61 | \( 1 - 8.56T + 61T^{2} \) |
| 67 | \( 1 + 4.85T + 67T^{2} \) |
| 71 | \( 1 + 5.32T + 71T^{2} \) |
| 73 | \( 1 - 7.70T + 73T^{2} \) |
| 79 | \( 1 - 11T + 79T^{2} \) |
| 83 | \( 1 + 7.47T + 83T^{2} \) |
| 89 | \( 1 + 3.76T + 89T^{2} \) |
| 97 | \( 1 - 1.14T + 97T^{2} \) |
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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
−10.78289067142785881077653490406, −9.863709486678904235572054203793, −9.396881642356561666798327339601, −8.313693641954583186886394218883, −7.08970735101694332655335795573, −6.05540157654284710868464828254, −5.03013832220991572063030127937, −4.08942123162730063370882791638, −2.14593612553279308138501529502, 0,
2.14593612553279308138501529502, 4.08942123162730063370882791638, 5.03013832220991572063030127937, 6.05540157654284710868464828254, 7.08970735101694332655335795573, 8.313693641954583186886394218883, 9.396881642356561666798327339601, 9.863709486678904235572054203793, 10.78289067142785881077653490406
