| L(s) = 1 | − 1.68·2-s + 1.38·3-s + 0.826·4-s + 5-s − 2.32·6-s + 1.97·8-s − 1.09·9-s − 1.68·10-s + 0.760·11-s + 1.14·12-s + 4.29·13-s + 1.38·15-s − 4.97·16-s − 2.73·17-s + 1.84·18-s − 0.807·19-s + 0.826·20-s − 1.27·22-s − 0.158·23-s + 2.72·24-s + 25-s − 7.21·26-s − 5.65·27-s − 3.69·29-s − 2.32·30-s − 31-s + 4.41·32-s + ⋯ |
| L(s) = 1 | − 1.18·2-s + 0.796·3-s + 0.413·4-s + 0.447·5-s − 0.947·6-s + 0.697·8-s − 0.365·9-s − 0.531·10-s + 0.229·11-s + 0.329·12-s + 1.18·13-s + 0.356·15-s − 1.24·16-s − 0.663·17-s + 0.433·18-s − 0.185·19-s + 0.184·20-s − 0.272·22-s − 0.0330·23-s + 0.555·24-s + 0.200·25-s − 1.41·26-s − 1.08·27-s − 0.686·29-s − 0.423·30-s − 0.179·31-s + 0.779·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7595 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 1.68T + 2T^{2} \) |
| 3 | \( 1 - 1.38T + 3T^{2} \) |
| 11 | \( 1 - 0.760T + 11T^{2} \) |
| 13 | \( 1 - 4.29T + 13T^{2} \) |
| 17 | \( 1 + 2.73T + 17T^{2} \) |
| 19 | \( 1 + 0.807T + 19T^{2} \) |
| 23 | \( 1 + 0.158T + 23T^{2} \) |
| 29 | \( 1 + 3.69T + 29T^{2} \) |
| 37 | \( 1 + 7.33T + 37T^{2} \) |
| 41 | \( 1 - 0.452T + 41T^{2} \) |
| 43 | \( 1 - 8.65T + 43T^{2} \) |
| 47 | \( 1 - 7.66T + 47T^{2} \) |
| 53 | \( 1 + 1.89T + 53T^{2} \) |
| 59 | \( 1 + 6.11T + 59T^{2} \) |
| 61 | \( 1 + 8.79T + 61T^{2} \) |
| 67 | \( 1 + 1.71T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 - 4.57T + 83T^{2} \) |
| 89 | \( 1 + 2.10T + 89T^{2} \) |
| 97 | \( 1 - 6.70T + 97T^{2} \) |
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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
−7.66306628735451351272249816472, −7.20820534102671512644292774127, −6.21343231089480463114521171773, −5.67546907254853837605406954538, −4.53818413027380002898082076739, −3.82884887920882232980933826841, −2.91050563395479275391556740327, −2.00495528113380146751248599137, −1.31552829483916580178198489008, 0,
1.31552829483916580178198489008, 2.00495528113380146751248599137, 2.91050563395479275391556740327, 3.82884887920882232980933826841, 4.53818413027380002898082076739, 5.67546907254853837605406954538, 6.21343231089480463114521171773, 7.20820534102671512644292774127, 7.66306628735451351272249816472
