| L(s) = 1 | − 2·7-s − 4·13-s + 7·17-s + 4·19-s − 9·23-s + 4·29-s + 7·31-s − 2·37-s − 2·41-s − 12·43-s + 9·47-s − 3·49-s − 13·53-s − 8·59-s + 5·61-s − 14·67-s − 8·71-s + 8·73-s − 9·79-s − 8·83-s + 6·89-s + 8·91-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 1.10·13-s + 1.69·17-s + 0.917·19-s − 1.87·23-s + 0.742·29-s + 1.25·31-s − 0.328·37-s − 0.312·41-s − 1.82·43-s + 1.31·47-s − 3/7·49-s − 1.78·53-s − 1.04·59-s + 0.640·61-s − 1.71·67-s − 0.949·71-s + 0.936·73-s − 1.01·79-s − 0.878·83-s + 0.635·89-s + 0.838·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.170156874\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.170156874\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 + 13 T + p T^{2} \) | 1.53.n |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 9 T + p T^{2} \) | 1.79.j |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−13.77569470998473, −13.21503005046624, −12.49435525559381, −12.21836809864011, −11.83972916771915, −11.49065543798459, −10.37811721713188, −10.17624519155346, −9.885375408105675, −9.408066506596754, −8.744468114813081, −7.989905494227604, −7.793381875084353, −7.249156798579891, −6.513990611682592, −6.166421138104214, −5.548515301414541, −5.006064269436602, −4.469460536021603, −3.744826979474827, −3.042004720552609, −2.905774647220245, −1.892357930067067, −1.271846806406697, −0.3418090711680404,
0.3418090711680404, 1.271846806406697, 1.892357930067067, 2.905774647220245, 3.042004720552609, 3.744826979474827, 4.469460536021603, 5.006064269436602, 5.548515301414541, 6.166421138104214, 6.513990611682592, 7.249156798579891, 7.793381875084353, 7.989905494227604, 8.744468114813081, 9.408066506596754, 9.885375408105675, 10.17624519155346, 10.37811721713188, 11.49065543798459, 11.83972916771915, 12.21836809864011, 12.49435525559381, 13.21503005046624, 13.77569470998473
