| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s + 4·11-s − 2·13-s + 16-s + 6·17-s + 4·19-s − 20-s − 4·22-s + 8·23-s + 25-s + 2·26-s + 2·29-s − 32-s − 6·34-s − 2·37-s − 4·38-s + 40-s + 6·41-s + 4·43-s + 4·44-s − 8·46-s + 8·47-s − 7·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s + 1.20·11-s − 0.554·13-s + 1/4·16-s + 1.45·17-s + 0.917·19-s − 0.223·20-s − 0.852·22-s + 1.66·23-s + 1/5·25-s + 0.392·26-s + 0.371·29-s − 0.176·32-s − 1.02·34-s − 0.328·37-s − 0.648·38-s + 0.158·40-s + 0.937·41-s + 0.609·43-s + 0.603·44-s − 1.17·46-s + 1.16·47-s − 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.632483199\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.632483199\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 79 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 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.i |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−7.72696844487441577222641498966, −7.45532857950574531214232241643, −6.75217246023286107067128865703, −5.93562736955905457222369280497, −5.16823904038290853830717495831, −4.30981564160990761292161411269, −3.36136456629968152124107448619, −2.81469278047054080275929072213, −1.45260254021803115867604290700, −0.806203820914913908526057743162,
0.806203820914913908526057743162, 1.45260254021803115867604290700, 2.81469278047054080275929072213, 3.36136456629968152124107448619, 4.30981564160990761292161411269, 5.16823904038290853830717495831, 5.93562736955905457222369280497, 6.75217246023286107067128865703, 7.45532857950574531214232241643, 7.72696844487441577222641498966
