| L(s) = 1 | − 2.34·2-s + 3-s + 3.48·4-s − 2.34·6-s + 7-s − 3.48·8-s + 9-s − 11-s + 3.48·12-s − 3.14·13-s − 2.34·14-s + 1.19·16-s − 6.48·17-s − 2.34·18-s + 7.17·19-s + 21-s + 2.34·22-s − 2.19·23-s − 3.48·24-s + 7.37·26-s + 27-s + 3.48·28-s + 0.949·29-s − 6.12·31-s + 4.17·32-s − 33-s + 15.2·34-s + ⋯ |
| L(s) = 1 | − 1.65·2-s + 0.577·3-s + 1.74·4-s − 0.956·6-s + 0.377·7-s − 1.23·8-s + 0.333·9-s − 0.301·11-s + 1.00·12-s − 0.872·13-s − 0.626·14-s + 0.299·16-s − 1.57·17-s − 0.552·18-s + 1.64·19-s + 0.218·21-s + 0.499·22-s − 0.458·23-s − 0.712·24-s + 1.44·26-s + 0.192·27-s + 0.659·28-s + 0.176·29-s − 1.10·31-s + 0.738·32-s − 0.174·33-s + 2.60·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 2.34T + 2T^{2} \) |
| 13 | \( 1 + 3.14T + 13T^{2} \) |
| 17 | \( 1 + 6.48T + 17T^{2} \) |
| 19 | \( 1 - 7.17T + 19T^{2} \) |
| 23 | \( 1 + 2.19T + 23T^{2} \) |
| 29 | \( 1 - 0.949T + 29T^{2} \) |
| 31 | \( 1 + 6.12T + 31T^{2} \) |
| 37 | \( 1 + 0.853T + 37T^{2} \) |
| 41 | \( 1 - 3.53T + 41T^{2} \) |
| 43 | \( 1 - 5.63T + 43T^{2} \) |
| 47 | \( 1 - 1.53T + 47T^{2} \) |
| 53 | \( 1 - 6.15T + 53T^{2} \) |
| 59 | \( 1 + 5.63T + 59T^{2} \) |
| 61 | \( 1 - 4.58T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 + 8.22T + 71T^{2} \) |
| 73 | \( 1 - 4.68T + 73T^{2} \) |
| 79 | \( 1 - 2.51T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + 18.3T + 89T^{2} \) |
| 97 | \( 1 - 5.00T + 97T^{2} \) |
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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
−7.86612797533956307934673363656, −7.28645034090434613565953653897, −6.89984671310668093064272349846, −5.76569368847602274726476154458, −4.86606406827879753705499959468, −3.95781939484852594426723740835, −2.71805845612083557651067597939, −2.19986628030637784003528770868, −1.22301040635593938212798527964, 0,
1.22301040635593938212798527964, 2.19986628030637784003528770868, 2.71805845612083557651067597939, 3.95781939484852594426723740835, 4.86606406827879753705499959468, 5.76569368847602274726476154458, 6.89984671310668093064272349846, 7.28645034090434613565953653897, 7.86612797533956307934673363656
