L(s) = 1 | − 1.73i·3-s + 3.74i·5-s − 2.99·9-s + 6.48·11-s + 6.48·15-s + 3.74i·17-s + 6.92i·19-s − 6.48·23-s − 9·25-s + 5.19i·27-s − 3.46i·31-s − 11.2i·33-s − 8·37-s − 3.74i·41-s − 11.2i·45-s + ⋯ |
L(s) = 1 | − 0.999i·3-s + 1.67i·5-s − 0.999·9-s + 1.95·11-s + 1.67·15-s + 0.907i·17-s + 1.58i·19-s − 1.35·23-s − 1.80·25-s + 0.999i·27-s − 0.622i·31-s − 1.95i·33-s − 1.31·37-s − 0.584i·41-s − 1.67i·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.417147706\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.417147706\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 3.74iT - 5T^{2} \) |
| 11 | \( 1 - 6.48T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 3.74iT - 17T^{2} \) |
| 19 | \( 1 - 6.92iT - 19T^{2} \) |
| 23 | \( 1 + 6.48T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 + 3.74iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 6.48T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 18.7iT - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−9.100678151218097145769525664544, −8.187853372965846273578686357856, −7.54862077601046934854046130560, −6.71454181234570373014540900870, −6.29706260960831750613270686979, −5.75108511108952738878256909680, −3.85710675981420994398508145602, −3.57551004297520296104969769845, −2.23624399118433364632000541698, −1.52049323604811346340513537758,
0.48232553811209032471025234907, 1.70430002482312423843975471442, 3.21533409493816358890282660618, 4.23988108097831991687896551062, 4.61877063343230696467379275239, 5.41961881032805411772378256416, 6.29291272895179281096972381301, 7.26637414522952538136769554892, 8.621100075088351538587801062873, 8.726579082038831138600136872055