| L(s) = 1 | − 2·5-s − 2·7-s + 4·11-s − 6·19-s + 4·23-s − 25-s + 8·29-s − 2·31-s + 4·35-s − 6·37-s + 6·41-s + 8·43-s − 8·47-s − 3·49-s − 12·53-s − 8·55-s − 4·59-s + 10·61-s − 2·67-s + 16·71-s − 14·73-s − 8·77-s + 4·79-s + 12·83-s − 6·89-s + 12·95-s + 10·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.755·7-s + 1.20·11-s − 1.37·19-s + 0.834·23-s − 1/5·25-s + 1.48·29-s − 0.359·31-s + 0.676·35-s − 0.986·37-s + 0.937·41-s + 1.21·43-s − 1.16·47-s − 3/7·49-s − 1.64·53-s − 1.07·55-s − 0.520·59-s + 1.28·61-s − 0.244·67-s + 1.89·71-s − 1.63·73-s − 0.911·77-s + 0.450·79-s + 1.31·83-s − 0.635·89-s + 1.23·95-s + 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.229047029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.229047029\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−15.54226230190403, −14.75515342765956, −14.49101511847307, −13.88341453741903, −13.07046738539403, −12.70357985481426, −12.14068911334570, −11.73375546345970, −10.93151441149859, −10.74225019316009, −9.783372493769215, −9.376341299720088, −8.742619185062257, −8.244289619091630, −7.607513521098715, −6.798126702982316, −6.538489791751226, −5.945591108284142, −4.916391620955505, −4.375418987037352, −3.742557331414418, −3.247356736224512, −2.390051244355129, −1.410849991952955, −0.4591744930409033,
0.4591744930409033, 1.410849991952955, 2.390051244355129, 3.247356736224512, 3.742557331414418, 4.375418987037352, 4.916391620955505, 5.945591108284142, 6.538489791751226, 6.798126702982316, 7.607513521098715, 8.244289619091630, 8.742619185062257, 9.376341299720088, 9.783372493769215, 10.74225019316009, 10.93151441149859, 11.73375546345970, 12.14068911334570, 12.70357985481426, 13.07046738539403, 13.88341453741903, 14.49101511847307, 14.75515342765956, 15.54226230190403
