| L(s) = 1 | + 5-s − 4·7-s − 11-s + 13-s − 4·17-s − 6·19-s + 2·23-s + 25-s − 6·29-s + 2·31-s − 4·35-s − 8·37-s − 2·41-s + 8·43-s + 9·49-s − 12·53-s − 55-s + 12·59-s − 14·61-s + 65-s − 16·67-s − 12·71-s + 4·73-s + 4·77-s − 16·79-s − 4·83-s − 4·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.51·7-s − 0.301·11-s + 0.277·13-s − 0.970·17-s − 1.37·19-s + 0.417·23-s + 1/5·25-s − 1.11·29-s + 0.359·31-s − 0.676·35-s − 1.31·37-s − 0.312·41-s + 1.21·43-s + 9/7·49-s − 1.64·53-s − 0.134·55-s + 1.56·59-s − 1.79·61-s + 0.124·65-s − 1.95·67-s − 1.42·71-s + 0.468·73-s + 0.455·77-s − 1.80·79-s − 0.439·83-s − 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.15716551960981, −13.40968049816174, −13.33255857163544, −12.81722710517758, −12.45866293517262, −11.86703212913690, −11.09162688489562, −10.77127726970140, −10.26367361504648, −9.840832472037086, −9.163301249789123, −8.896404440864010, −8.477105896839558, −7.545419852357626, −7.163842153036237, −6.529709699390730, −6.157418827118961, −5.801448639296050, −4.993206454445877, −4.393029540448828, −3.835304494079446, −3.166564764315996, −2.681408117464016, −2.028201338444723, −1.327414136740375, 0, 0,
1.327414136740375, 2.028201338444723, 2.681408117464016, 3.166564764315996, 3.835304494079446, 4.393029540448828, 4.993206454445877, 5.801448639296050, 6.157418827118961, 6.529709699390730, 7.163842153036237, 7.545419852357626, 8.477105896839558, 8.896404440864010, 9.163301249789123, 9.840832472037086, 10.26367361504648, 10.77127726970140, 11.09162688489562, 11.86703212913690, 12.45866293517262, 12.81722710517758, 13.33255857163544, 13.40968049816174, 14.15716551960981
