L(s) = 1 | + 3-s + 7-s + 9-s + 11-s + 6·13-s + 2·17-s − 4·19-s + 21-s + 8·23-s + 27-s − 2·29-s + 33-s + 10·37-s + 6·39-s + 2·41-s + 8·43-s + 49-s + 2·51-s + 10·53-s − 4·57-s − 4·59-s + 2·61-s + 63-s + 12·67-s + 8·69-s − 4·71-s − 6·73-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s + 1.66·13-s + 0.485·17-s − 0.917·19-s + 0.218·21-s + 1.66·23-s + 0.192·27-s − 0.371·29-s + 0.174·33-s + 1.64·37-s + 0.960·39-s + 0.312·41-s + 1.21·43-s + 1/7·49-s + 0.280·51-s + 1.37·53-s − 0.529·57-s − 0.520·59-s + 0.256·61-s + 0.125·63-s + 1.46·67-s + 0.963·69-s − 0.474·71-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.245312721\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.245312721\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−13.77788838587282, −13.35444946992905, −12.96328721032771, −12.52730618901565, −11.82533011180075, −11.26572032531040, −10.81435305413699, −10.62212636623754, −9.688747574253677, −9.315972425528891, −8.712844518134205, −8.497732105421528, −7.821666963432197, −7.367401033270042, −6.734552844115751, −6.144744561091882, −5.756114353403596, −4.968422677939860, −4.388071017168300, −3.809478825139900, −3.396058726014361, −2.604329487558144, −2.062904630100040, −1.130423059577806, −0.8433415912489656,
0.8433415912489656, 1.130423059577806, 2.062904630100040, 2.604329487558144, 3.396058726014361, 3.809478825139900, 4.388071017168300, 4.968422677939860, 5.756114353403596, 6.144744561091882, 6.734552844115751, 7.367401033270042, 7.821666963432197, 8.497732105421528, 8.712844518134205, 9.315972425528891, 9.688747574253677, 10.62212636623754, 10.81435305413699, 11.26572032531040, 11.82533011180075, 12.52730618901565, 12.96328721032771, 13.35444946992905, 13.77788838587282