| L(s) = 1 | − 414. i·5-s + (1.95e3 + 1.39e3i)7-s − 2.15e4·11-s − 4.37e3i·13-s + 1.16e5i·17-s − 1.90e5i·19-s + 1.25e5·23-s + 2.19e5·25-s + 4.83e5·29-s − 2.45e5i·31-s + (5.76e5 − 8.09e5i)35-s − 1.09e5·37-s + 3.89e6i·41-s − 1.39e6·43-s + 1.96e6i·47-s + ⋯ |
| L(s) = 1 | − 0.662i·5-s + (0.814 + 0.579i)7-s − 1.47·11-s − 0.153i·13-s + 1.40i·17-s − 1.46i·19-s + 0.448·23-s + 0.561·25-s + 0.684·29-s − 0.265i·31-s + (0.384 − 0.539i)35-s − 0.0584·37-s + 1.37i·41-s − 0.408·43-s + 0.401i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.814 + 0.579i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.814 + 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.041703187\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.041703187\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.95e3 - 1.39e3i)T \) |
| good | 5 | \( 1 + 414. iT - 3.90e5T^{2} \) |
| 11 | \( 1 + 2.15e4T + 2.14e8T^{2} \) |
| 13 | \( 1 + 4.37e3iT - 8.15e8T^{2} \) |
| 17 | \( 1 - 1.16e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 1.90e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 1.25e5T + 7.83e10T^{2} \) |
| 29 | \( 1 - 4.83e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + 2.45e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 + 1.09e5T + 3.51e12T^{2} \) |
| 41 | \( 1 - 3.89e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 1.39e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 1.96e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 7.39e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + 1.79e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 2.31e7iT - 1.91e14T^{2} \) |
| 67 | \( 1 - 7.63e6T + 4.06e14T^{2} \) |
| 71 | \( 1 - 3.98e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + 2.57e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 - 2.07e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 6.12e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 7.65e6iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 7.16e7iT - 7.83e15T^{2} \) |
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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
−10.69252336075574875488229768820, −9.459339791130784728602527948810, −8.396924152252963948581207367918, −7.938292155192991155631916639060, −6.45704323896549081746949297183, −5.17837106429454271017586779709, −4.71076629462112854997077671852, −2.98882112626654435550493347761, −1.87809751614725136718164038542, −0.58885893747565996288010273612,
0.797431757855628430556781859612, 2.21963493415184230370301436523, 3.27404941966337661007364280617, 4.66154352766798934201372517547, 5.54610328457619728369793691661, 6.99837508098775460212356657055, 7.65204751745258912153976620471, 8.630119791526002389222184007284, 10.08006579148348142338770644906, 10.60610427171230899826676130948
