| L(s) = 1 | − 1.30i·3-s − i·5-s − 1.10·7-s + 1.30·9-s + 2.46·11-s − 6.60·13-s − 1.30·15-s − 6.27i·17-s + 2.35·19-s + 1.43i·21-s + (−3.96 + 2.70i)23-s − 25-s − 5.60i·27-s − 6.89·29-s + 3.52i·31-s + ⋯ |
| L(s) = 1 | − 0.750i·3-s − 0.447i·5-s − 0.418·7-s + 0.436·9-s + 0.743·11-s − 1.83·13-s − 0.335·15-s − 1.52i·17-s + 0.540·19-s + 0.313i·21-s + (−0.825 + 0.563i)23-s − 0.200·25-s − 1.07i·27-s − 1.28·29-s + 0.632i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8641000627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8641000627\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 23 | \( 1 + (3.96 - 2.70i)T \) |
| good | 3 | \( 1 + 1.30iT - 3T^{2} \) |
| 7 | \( 1 + 1.10T + 7T^{2} \) |
| 11 | \( 1 - 2.46T + 11T^{2} \) |
| 13 | \( 1 + 6.60T + 13T^{2} \) |
| 17 | \( 1 + 6.27iT - 17T^{2} \) |
| 19 | \( 1 - 2.35T + 19T^{2} \) |
| 29 | \( 1 + 6.89T + 29T^{2} \) |
| 31 | \( 1 - 3.52iT - 31T^{2} \) |
| 37 | \( 1 + 2.61iT - 37T^{2} \) |
| 41 | \( 1 + 3.37T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 + 6.35iT - 47T^{2} \) |
| 53 | \( 1 + 10.0iT - 53T^{2} \) |
| 59 | \( 1 + 3.46iT - 59T^{2} \) |
| 61 | \( 1 - 3.82iT - 61T^{2} \) |
| 67 | \( 1 + 15.4T + 67T^{2} \) |
| 71 | \( 1 - 8.95iT - 71T^{2} \) |
| 73 | \( 1 + 0.691T + 73T^{2} \) |
| 79 | \( 1 - 5.32T + 79T^{2} \) |
| 83 | \( 1 + 4.45T + 83T^{2} \) |
| 89 | \( 1 + 4.50iT - 89T^{2} \) |
| 97 | \( 1 - 5.76iT - 97T^{2} \) |
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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
−9.035411524870032965990439154055, −7.78431335121019132893114546499, −7.28764249124413712723133257842, −6.75483531252419329580526811364, −5.59970323960250446437119752745, −4.85389785666410602578078638537, −3.85737930928094337113058583733, −2.63831027296631400772961532856, −1.63990229010165994714299183061, −0.30492811049263686626538770173,
1.74932411151232598678635514265, 2.93113466225852192708796329788, 3.97206641510072171978755841871, 4.48699109758352276689392089212, 5.64104554238360073179394277906, 6.42577536956325899541139241053, 7.32382726040798566366561067578, 7.929141978792539944166315628694, 9.288043417565172738484124323310, 9.523398596264053891156147415711
