| L(s) = 1 | + 3-s − 5-s + 9-s − 11-s − 2·13-s − 15-s − 2·17-s + 25-s + 27-s + 6·29-s + 4·31-s − 33-s + 6·37-s − 2·39-s − 2·41-s − 8·43-s − 45-s − 7·49-s − 2·51-s + 10·53-s + 55-s + 12·59-s − 2·61-s + 2·65-s + 4·67-s + 8·71-s − 6·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 0.258·15-s − 0.485·17-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.718·31-s − 0.174·33-s + 0.986·37-s − 0.320·39-s − 0.312·41-s − 1.21·43-s − 0.149·45-s − 49-s − 0.280·51-s + 1.37·53-s + 0.134·55-s + 1.56·59-s − 0.256·61-s + 0.248·65-s + 0.488·67-s + 0.949·71-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.994108258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.994108258\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−8.361085748000281903855047229956, −7.51999851021943740418772301493, −6.88805534128100675052994851129, −6.17819049312011993114639959501, −5.06432437932069328474822692369, −4.53762606893164222582530885715, −3.64456472172663735139724854519, −2.82690070684511436559084244196, −2.06997285225943099647389382381, −0.72993655247130672033335741955,
0.72993655247130672033335741955, 2.06997285225943099647389382381, 2.82690070684511436559084244196, 3.64456472172663735139724854519, 4.53762606893164222582530885715, 5.06432437932069328474822692369, 6.17819049312011993114639959501, 6.88805534128100675052994851129, 7.51999851021943740418772301493, 8.361085748000281903855047229956
