| L(s) = 1 | + 3-s + 5-s + 3·7-s + 9-s − 3·11-s + 15-s + 17-s + 8·19-s + 3·21-s − 3·23-s + 25-s + 27-s + 6·29-s + 10·31-s − 3·33-s + 3·35-s + 37-s − 5·41-s − 6·43-s + 45-s + 8·47-s + 2·49-s + 51-s + 5·53-s − 3·55-s + 8·57-s − 4·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.13·7-s + 1/3·9-s − 0.904·11-s + 0.258·15-s + 0.242·17-s + 1.83·19-s + 0.654·21-s − 0.625·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 1.79·31-s − 0.522·33-s + 0.507·35-s + 0.164·37-s − 0.780·41-s − 0.914·43-s + 0.149·45-s + 1.16·47-s + 2/7·49-s + 0.140·51-s + 0.686·53-s − 0.404·55-s + 1.05·57-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.133183346\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.133183346\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 11 T + p T^{2} \) | 1.71.l |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−15.66453834369905, −15.10643191922906, −14.40620108365117, −13.94276104231642, −13.70039222273340, −13.09616078111563, −12.28753765991998, −11.75875920807503, −11.39584436418838, −10.39699170179414, −10.14328065543711, −9.637609668701142, −8.768081882318005, −8.256226408841093, −7.880733398695307, −7.261856640768970, −6.563525444099565, −5.688637141514115, −5.135761049856115, −4.689687283743327, −3.825086377954130, −2.920887420538031, −2.486056218481606, −1.551281408969946, −0.8692431636913283,
0.8692431636913283, 1.551281408969946, 2.486056218481606, 2.920887420538031, 3.825086377954130, 4.689687283743327, 5.135761049856115, 5.688637141514115, 6.563525444099565, 7.261856640768970, 7.880733398695307, 8.256226408841093, 8.768081882318005, 9.637609668701142, 10.14328065543711, 10.39699170179414, 11.39584436418838, 11.75875920807503, 12.28753765991998, 13.09616078111563, 13.70039222273340, 13.94276104231642, 14.40620108365117, 15.10643191922906, 15.66453834369905
