| L(s) = 1 | − 2·4-s + 3.46·5-s − 7-s + 3.46·11-s − 13-s + 4·16-s − 3.46·17-s + 5·19-s − 6.92·20-s + 6.99·25-s + 2·28-s + 31-s − 3.46·35-s + 5·37-s + 3.46·41-s + 8·43-s − 6.92·44-s + 10.3·47-s − 6·49-s + 2·52-s + 3.46·53-s + 11.9·55-s − 13.8·59-s + 5·61-s − 8·64-s − 3.46·65-s + 5·67-s + ⋯ |
| L(s) = 1 | − 4-s + 1.54·5-s − 0.377·7-s + 1.04·11-s − 0.277·13-s + 16-s − 0.840·17-s + 1.14·19-s − 1.54·20-s + 1.39·25-s + 0.377·28-s + 0.179·31-s − 0.585·35-s + 0.821·37-s + 0.541·41-s + 1.21·43-s − 1.04·44-s + 1.51·47-s − 0.857·49-s + 0.277·52-s + 0.475·53-s + 1.61·55-s − 1.80·59-s + 0.640·61-s − 64-s − 0.429·65-s + 0.610·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.651750442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.651750442\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 5 | \( 1 - 3.46T + 5T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 - 3.46T + 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 - 5T + 67T^{2} \) |
| 71 | \( 1 + 6.92T + 71T^{2} \) |
| 73 | \( 1 - 11T + 73T^{2} \) |
| 79 | \( 1 + T + 79T^{2} \) |
| 83 | \( 1 - 3.46T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 5T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.911809846592584450323299545132, −9.277917035403339643756163323383, −9.030656670531920628561741815314, −7.67444565112721543409497728603, −6.51178611587823318733120043472, −5.82996264208230393975028440681, −4.93460111281073043164564862460, −3.89853889980200916525998639103, −2.56505452685828550606153552642, −1.14630803571439896169326558963,
1.14630803571439896169326558963, 2.56505452685828550606153552642, 3.89853889980200916525998639103, 4.93460111281073043164564862460, 5.82996264208230393975028440681, 6.51178611587823318733120043472, 7.67444565112721543409497728603, 9.030656670531920628561741815314, 9.277917035403339643756163323383, 9.911809846592584450323299545132
