| L(s) = 1 | − 7-s − 2·11-s + 5·13-s − 4·17-s + 5·19-s + 2·23-s − 10·29-s + 8·31-s + 3·37-s − 6·41-s + 4·43-s + 8·47-s − 6·49-s + 6·53-s − 4·59-s − 5·61-s − 7·67-s + 6·71-s + 9·73-s + 2·77-s − 3·79-s − 2·83-s − 5·91-s − 7·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 0.603·11-s + 1.38·13-s − 0.970·17-s + 1.14·19-s + 0.417·23-s − 1.85·29-s + 1.43·31-s + 0.493·37-s − 0.937·41-s + 0.609·43-s + 1.16·47-s − 6/7·49-s + 0.824·53-s − 0.520·59-s − 0.640·61-s − 0.855·67-s + 0.712·71-s + 1.05·73-s + 0.227·77-s − 0.337·79-s − 0.219·83-s − 0.524·91-s − 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.876681008\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.876681008\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 + 3 T + p T^{2} \) | 1.79.d |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−16.57949714172863, −15.81558107087056, −15.45516452874188, −15.08413874822433, −13.93831180503012, −13.74639245404357, −13.08792874264115, −12.70846314904655, −11.76150788221996, −11.31866229325861, −10.76250023573060, −10.14921634626640, −9.375846589685757, −8.950463008209812, −8.225655750554436, −7.589291926220532, −6.921740823993759, −6.190332085199078, −5.665364537778802, −4.892427616081506, −4.068652912586691, −3.372821969704327, −2.658986271692527, −1.659430534106384, −0.6476249905686665,
0.6476249905686665, 1.659430534106384, 2.658986271692527, 3.372821969704327, 4.068652912586691, 4.892427616081506, 5.665364537778802, 6.190332085199078, 6.921740823993759, 7.589291926220532, 8.225655750554436, 8.950463008209812, 9.375846589685757, 10.14921634626640, 10.76250023573060, 11.31866229325861, 11.76150788221996, 12.70846314904655, 13.08792874264115, 13.74639245404357, 13.93831180503012, 15.08413874822433, 15.45516452874188, 15.81558107087056, 16.57949714172863
