| L(s) = 1 | − 2.26·2-s − 3-s + 3.13·4-s + 2.26·6-s − 0.159·7-s − 2.58·8-s + 9-s − 3.24·11-s − 3.13·12-s − 7.05·13-s + 0.362·14-s − 0.424·16-s − 7.63·17-s − 2.26·18-s − 3.99·19-s + 0.159·21-s + 7.36·22-s − 2.06·23-s + 2.58·24-s + 15.9·26-s − 27-s − 0.501·28-s − 29-s + 10.2·31-s + 6.12·32-s + 3.24·33-s + 17.3·34-s + ⋯ |
| L(s) = 1 | − 1.60·2-s − 0.577·3-s + 1.56·4-s + 0.925·6-s − 0.0604·7-s − 0.912·8-s + 0.333·9-s − 0.979·11-s − 0.906·12-s − 1.95·13-s + 0.0968·14-s − 0.106·16-s − 1.85·17-s − 0.534·18-s − 0.915·19-s + 0.0348·21-s + 1.57·22-s − 0.430·23-s + 0.527·24-s + 3.13·26-s − 0.192·27-s − 0.0948·28-s − 0.185·29-s + 1.83·31-s + 1.08·32-s + 0.565·33-s + 2.96·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1832107731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1832107731\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 2.26T + 2T^{2} \) |
| 7 | \( 1 + 0.159T + 7T^{2} \) |
| 11 | \( 1 + 3.24T + 11T^{2} \) |
| 13 | \( 1 + 7.05T + 13T^{2} \) |
| 17 | \( 1 + 7.63T + 17T^{2} \) |
| 19 | \( 1 + 3.99T + 19T^{2} \) |
| 23 | \( 1 + 2.06T + 23T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 - 4.27T + 37T^{2} \) |
| 41 | \( 1 + 5.12T + 41T^{2} \) |
| 43 | \( 1 - 3.94T + 43T^{2} \) |
| 47 | \( 1 + 5.61T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 + 7.06T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 - 14.9T + 67T^{2} \) |
| 71 | \( 1 - 3.43T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 - 2.59T + 83T^{2} \) |
| 89 | \( 1 - 4.33T + 89T^{2} \) |
| 97 | \( 1 - 3.88T + 97T^{2} \) |
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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
−9.181754280083428117831055275504, −8.191425241229973857971680899807, −7.80764020809839228292300294980, −6.77365005812326555602555547095, −6.44153623047584144636398724815, −5.00115013448802258041752538765, −4.48596485724729850408283162368, −2.61023099584197387327062384380, −2.03738678077670244844773244886, −0.34138934485228128137122512189,
0.34138934485228128137122512189, 2.03738678077670244844773244886, 2.61023099584197387327062384380, 4.48596485724729850408283162368, 5.00115013448802258041752538765, 6.44153623047584144636398724815, 6.77365005812326555602555547095, 7.80764020809839228292300294980, 8.191425241229973857971680899807, 9.181754280083428117831055275504
