| L(s) = 1 | − 2-s + 4-s + 3·5-s + 4·7-s − 8-s − 3·10-s + 13-s − 4·14-s + 16-s − 3·17-s + 4·19-s + 3·20-s + 4·25-s − 26-s + 4·28-s + 9·29-s − 4·31-s − 32-s + 3·34-s + 12·35-s − 37-s − 4·38-s − 3·40-s + 6·41-s − 8·43-s + 12·47-s + 9·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.34·5-s + 1.51·7-s − 0.353·8-s − 0.948·10-s + 0.277·13-s − 1.06·14-s + 1/4·16-s − 0.727·17-s + 0.917·19-s + 0.670·20-s + 4/5·25-s − 0.196·26-s + 0.755·28-s + 1.67·29-s − 0.718·31-s − 0.176·32-s + 0.514·34-s + 2.02·35-s − 0.164·37-s − 0.648·38-s − 0.474·40-s + 0.937·41-s − 1.21·43-s + 1.75·47-s + 9/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19602 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19602 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.060593492\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.060593492\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−15.65597630587806, −15.23016556917662, −14.50284837301708, −14.01733397890492, −13.71568821122412, −13.05096012380496, −12.21868497845610, −11.75953436967111, −11.07123165171086, −10.69970379187588, −10.08356491436524, −9.574239749431673, −8.789334258951279, −8.621262346420856, −7.787703194398690, −7.249223268602146, −6.530276199961455, −5.895166097956818, −5.284501482580921, −4.770552288191000, −3.889926445567422, −2.760204378497833, −2.185309175637112, −1.517564371792932, −0.8750453045688398,
0.8750453045688398, 1.517564371792932, 2.185309175637112, 2.760204378497833, 3.889926445567422, 4.770552288191000, 5.284501482580921, 5.895166097956818, 6.530276199961455, 7.249223268602146, 7.787703194398690, 8.621262346420856, 8.789334258951279, 9.574239749431673, 10.08356491436524, 10.69970379187588, 11.07123165171086, 11.75953436967111, 12.21868497845610, 13.05096012380496, 13.71568821122412, 14.01733397890492, 14.50284837301708, 15.23016556917662, 15.65597630587806
