| L(s) = 1 | − 2-s − 3-s − 5-s + 6-s + 8-s + 9-s + 10-s + 15-s − 16-s − 17-s − 18-s − 2·19-s + 23-s − 24-s + 25-s − 27-s − 30-s − 31-s + 34-s + 2·38-s − 40-s − 45-s − 46-s + 47-s + 48-s + 49-s − 50-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s − 5-s + 6-s + 8-s + 9-s + 10-s + 15-s − 16-s − 17-s − 18-s − 2·19-s + 23-s − 24-s + 25-s − 27-s − 30-s − 31-s + 34-s + 2·38-s − 40-s − 45-s − 46-s + 47-s + 48-s + 49-s − 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2731264667\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2731264667\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + T + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( ( 1 + T )^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( ( 1 - T )^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−9.270015142252286647298766835934, −8.763686150174420374412711117953, −7.966909686500984955965928063521, −7.09203726797406700849896031030, −6.62891197057680533712763043815, −5.35795294236396707792593092030, −4.44458075485389115765399121033, −3.94173941702451812884082078010, −2.11224987530574944279317054000, −0.63435997981231660576126859035,
0.63435997981231660576126859035, 2.11224987530574944279317054000, 3.94173941702451812884082078010, 4.44458075485389115765399121033, 5.35795294236396707792593092030, 6.62891197057680533712763043815, 7.09203726797406700849896031030, 7.966909686500984955965928063521, 8.763686150174420374412711117953, 9.270015142252286647298766835934
