| L(s) = 1 | + 2·5-s + 2·7-s + 4·11-s + 13-s + 2·19-s − 4·23-s − 25-s + 2·31-s + 4·35-s + 10·37-s − 2·41-s − 8·43-s − 3·49-s + 12·53-s + 8·55-s + 12·59-s + 6·61-s + 2·65-s + 6·67-s + 8·71-s − 2·73-s + 8·77-s + 12·79-s − 4·83-s − 14·89-s + 2·91-s + 4·95-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s + 1.20·11-s + 0.277·13-s + 0.458·19-s − 0.834·23-s − 1/5·25-s + 0.359·31-s + 0.676·35-s + 1.64·37-s − 0.312·41-s − 1.21·43-s − 3/7·49-s + 1.64·53-s + 1.07·55-s + 1.56·59-s + 0.768·61-s + 0.248·65-s + 0.733·67-s + 0.949·71-s − 0.234·73-s + 0.911·77-s + 1.35·79-s − 0.439·83-s − 1.48·89-s + 0.209·91-s + 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.191978390\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.191978390\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 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 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−8.110978655637205218383153371196, −7.04443061267946486481986390805, −6.50148363870543139899229499703, −5.77230682133162482102185007896, −5.20926919989103888445718632029, −4.26311349156385474544953922610, −3.68379839020729168882521783347, −2.50387478648239223489768216370, −1.75759038819220582302893495453, −0.967338466071760646824971885007,
0.967338466071760646824971885007, 1.75759038819220582302893495453, 2.50387478648239223489768216370, 3.68379839020729168882521783347, 4.26311349156385474544953922610, 5.20926919989103888445718632029, 5.77230682133162482102185007896, 6.50148363870543139899229499703, 7.04443061267946486481986390805, 8.110978655637205218383153371196
