L(s) = 1 | − 2.36·3-s − 5-s + 7-s + 2.60·9-s + 5.04·11-s + 13-s + 2.36·15-s + 1.60·17-s − 3.29·19-s − 2.36·21-s + 2.31·23-s + 25-s + 0.928·27-s + 1.32·29-s − 4.36·31-s − 11.9·33-s − 35-s + 10.6·37-s − 2.36·39-s + 0.249·41-s + 3.68·43-s − 2.60·45-s − 3.21·47-s + 49-s − 3.80·51-s − 4.73·53-s − 5.04·55-s + ⋯ |
L(s) = 1 | − 1.36·3-s − 0.447·5-s + 0.377·7-s + 0.869·9-s + 1.52·11-s + 0.277·13-s + 0.611·15-s + 0.389·17-s − 0.756·19-s − 0.516·21-s + 0.481·23-s + 0.200·25-s + 0.178·27-s + 0.245·29-s − 0.784·31-s − 2.08·33-s − 0.169·35-s + 1.74·37-s − 0.379·39-s + 0.0390·41-s + 0.562·43-s − 0.388·45-s − 0.469·47-s + 0.142·49-s − 0.533·51-s − 0.650·53-s − 0.680·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.136051375\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.136051375\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 + 2.36T + 3T^{2} \) |
| 11 | \( 1 - 5.04T + 11T^{2} \) |
| 17 | \( 1 - 1.60T + 17T^{2} \) |
| 19 | \( 1 + 3.29T + 19T^{2} \) |
| 23 | \( 1 - 2.31T + 23T^{2} \) |
| 29 | \( 1 - 1.32T + 29T^{2} \) |
| 31 | \( 1 + 4.36T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 - 0.249T + 41T^{2} \) |
| 43 | \( 1 - 3.68T + 43T^{2} \) |
| 47 | \( 1 + 3.21T + 47T^{2} \) |
| 53 | \( 1 + 4.73T + 53T^{2} \) |
| 59 | \( 1 + 5.58T + 59T^{2} \) |
| 61 | \( 1 + 0.142T + 61T^{2} \) |
| 67 | \( 1 + 6.34T + 67T^{2} \) |
| 71 | \( 1 + 6.42T + 71T^{2} \) |
| 73 | \( 1 - 3.35T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 + 4.73T + 83T^{2} \) |
| 89 | \( 1 + 7.27T + 89T^{2} \) |
| 97 | \( 1 - 3.52T + 97T^{2} \) |
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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
−8.595823327269010297221524725790, −7.65449972395921680861396240594, −6.90478664043610417092164117916, −6.20400152426156582583501758008, −5.72336147666840447301418170827, −4.62729753820452156260178800075, −4.23432456636175315536530029595, −3.15035568591156924179030359768, −1.63299128221074507844215043857, −0.71108026784411981551783204096,
0.71108026784411981551783204096, 1.63299128221074507844215043857, 3.15035568591156924179030359768, 4.23432456636175315536530029595, 4.62729753820452156260178800075, 5.72336147666840447301418170827, 6.20400152426156582583501758008, 6.90478664043610417092164117916, 7.65449972395921680861396240594, 8.595823327269010297221524725790