L(s) = 1 | + 3.39·3-s − 1.81·5-s + 1.32·7-s + 8.51·9-s + 11-s + 3.18·13-s − 6.16·15-s + 7.48·17-s + 5.16·19-s + 4.49·21-s − 6.82·23-s − 1.70·25-s + 18.7·27-s + 4.49·29-s − 2.24·31-s + 3.39·33-s − 2.40·35-s − 1.13·37-s + 10.7·39-s + 7.09·41-s − 12.1·43-s − 15.4·45-s − 4.76·47-s − 5.24·49-s + 25.4·51-s − 13.9·53-s − 1.81·55-s + ⋯ |
L(s) = 1 | + 1.95·3-s − 0.812·5-s + 0.500·7-s + 2.83·9-s + 0.301·11-s + 0.882·13-s − 1.59·15-s + 1.81·17-s + 1.18·19-s + 0.980·21-s − 1.42·23-s − 0.340·25-s + 3.60·27-s + 0.834·29-s − 0.402·31-s + 0.590·33-s − 0.406·35-s − 0.185·37-s + 1.72·39-s + 1.10·41-s − 1.85·43-s − 2.30·45-s − 0.695·47-s − 0.749·49-s + 3.55·51-s − 1.91·53-s − 0.244·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.757243748\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.757243748\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 137 | \( 1 - T \) |
good | 3 | \( 1 - 3.39T + 3T^{2} \) |
| 5 | \( 1 + 1.81T + 5T^{2} \) |
| 7 | \( 1 - 1.32T + 7T^{2} \) |
| 13 | \( 1 - 3.18T + 13T^{2} \) |
| 17 | \( 1 - 7.48T + 17T^{2} \) |
| 19 | \( 1 - 5.16T + 19T^{2} \) |
| 23 | \( 1 + 6.82T + 23T^{2} \) |
| 29 | \( 1 - 4.49T + 29T^{2} \) |
| 31 | \( 1 + 2.24T + 31T^{2} \) |
| 37 | \( 1 + 1.13T + 37T^{2} \) |
| 41 | \( 1 - 7.09T + 41T^{2} \) |
| 43 | \( 1 + 12.1T + 43T^{2} \) |
| 47 | \( 1 + 4.76T + 47T^{2} \) |
| 53 | \( 1 + 13.9T + 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 + 1.73T + 61T^{2} \) |
| 67 | \( 1 + 4.55T + 67T^{2} \) |
| 71 | \( 1 + 3.68T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 + 7.47T + 83T^{2} \) |
| 89 | \( 1 - 2.74T + 89T^{2} \) |
| 97 | \( 1 - 10.3T + 97T^{2} \) |
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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
−8.106895871535124628449151164462, −7.73305021588872781121482895818, −7.05063870665874562223903701166, −6.03333627553385182281842892276, −4.93028132427298699111699393808, −4.09991913273467508577675361016, −3.44682064000212845027309088237, −3.12344373097424576449597972585, −1.83950766613377255324986436771, −1.18597025838020326458203315946,
1.18597025838020326458203315946, 1.83950766613377255324986436771, 3.12344373097424576449597972585, 3.44682064000212845027309088237, 4.09991913273467508577675361016, 4.93028132427298699111699393808, 6.03333627553385182281842892276, 7.05063870665874562223903701166, 7.73305021588872781121482895818, 8.106895871535124628449151164462