| L(s) = 1 | − 3·5-s + 3·7-s − 4·11-s + 13-s − 4·17-s − 19-s − 4·23-s + 4·25-s − 10·31-s − 9·35-s + 5·37-s − 7·41-s + 8·43-s − 2·47-s + 2·49-s + 6·53-s + 12·55-s + 10·59-s + 2·61-s − 3·65-s − 4·67-s + 13·71-s + 15·73-s − 12·77-s − 2·79-s + 13·83-s + 12·85-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 1.13·7-s − 1.20·11-s + 0.277·13-s − 0.970·17-s − 0.229·19-s − 0.834·23-s + 4/5·25-s − 1.79·31-s − 1.52·35-s + 0.821·37-s − 1.09·41-s + 1.21·43-s − 0.291·47-s + 2/7·49-s + 0.824·53-s + 1.61·55-s + 1.30·59-s + 0.256·61-s − 0.372·65-s − 0.488·67-s + 1.54·71-s + 1.75·73-s − 1.36·77-s − 0.225·79-s + 1.42·83-s + 1.30·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.004482692\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.004482692\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 13 T + p T^{2} \) | 1.71.an |
| 73 | \( 1 - 15 T + p T^{2} \) | 1.73.ap |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 - 13 T + p T^{2} \) | 1.83.an |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−7.942686672713278059624793165756, −7.33190795172537514479883803747, −6.59048067279949816488608563279, −5.50206877930979521961810516088, −5.02945085223159448371343331136, −4.14185457532826496865186803931, −3.77802169035175164144725969419, −2.59115781866271828738401926429, −1.84734244626117388227365069350, −0.47833317011550482096195345405,
0.47833317011550482096195345405, 1.84734244626117388227365069350, 2.59115781866271828738401926429, 3.77802169035175164144725969419, 4.14185457532826496865186803931, 5.02945085223159448371343331136, 5.50206877930979521961810516088, 6.59048067279949816488608563279, 7.33190795172537514479883803747, 7.942686672713278059624793165756
