| L(s) = 1 | − 2·7-s + 6·11-s + 4·13-s + 17-s + 2·19-s − 9·23-s − 6·29-s + 10·31-s − 5·37-s − 3·41-s + 2·43-s − 6·47-s − 3·49-s + 9·53-s − 3·59-s + 7·61-s + 14·67-s + 15·71-s + 8·73-s − 12·77-s − 14·79-s − 3·83-s − 8·91-s − 16·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 1.80·11-s + 1.10·13-s + 0.242·17-s + 0.458·19-s − 1.87·23-s − 1.11·29-s + 1.79·31-s − 0.821·37-s − 0.468·41-s + 0.304·43-s − 0.875·47-s − 3/7·49-s + 1.23·53-s − 0.390·59-s + 0.896·61-s + 1.71·67-s + 1.78·71-s + 0.936·73-s − 1.36·77-s − 1.57·79-s − 0.329·83-s − 0.838·91-s − 1.62·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 244800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 244800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.675418647\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.675418647\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−12.77374231235439, −12.37276023569572, −11.95624820259705, −11.40089977982212, −11.31252202500447, −10.45759055399617, −9.903410882594581, −9.728300024633802, −9.208912391700106, −8.671975540820368, −8.159519847923789, −7.894131809470739, −6.870210364895410, −6.699984469056470, −6.360165058845518, −5.681311622612458, −5.399410145941555, −4.432108011465900, −3.926697280967083, −3.699009989963879, −3.176550471569608, −2.359285701369835, −1.684463069779106, −1.185048055577711, −0.4782706746880503,
0.4782706746880503, 1.185048055577711, 1.684463069779106, 2.359285701369835, 3.176550471569608, 3.699009989963879, 3.926697280967083, 4.432108011465900, 5.399410145941555, 5.681311622612458, 6.360165058845518, 6.699984469056470, 6.870210364895410, 7.894131809470739, 8.159519847923789, 8.671975540820368, 9.208912391700106, 9.728300024633802, 9.903410882594581, 10.45759055399617, 11.31252202500447, 11.40089977982212, 11.95624820259705, 12.37276023569572, 12.77374231235439
