| L(s) = 1 | − 2-s + 4-s + 5-s − 5·7-s − 8-s − 10-s + 4·11-s − 13-s + 5·14-s + 16-s + 3·17-s + 19-s + 20-s − 4·22-s − 7·23-s + 25-s + 26-s − 5·28-s + 3·29-s − 2·31-s − 32-s − 3·34-s − 5·35-s − 2·37-s − 38-s − 40-s + 6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.88·7-s − 0.353·8-s − 0.316·10-s + 1.20·11-s − 0.277·13-s + 1.33·14-s + 1/4·16-s + 0.727·17-s + 0.229·19-s + 0.223·20-s − 0.852·22-s − 1.45·23-s + 1/5·25-s + 0.196·26-s − 0.944·28-s + 0.557·29-s − 0.359·31-s − 0.176·32-s − 0.514·34-s − 0.845·35-s − 0.328·37-s − 0.162·38-s − 0.158·40-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.044058985\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.044058985\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 19 | \( 1 - T \) | |
| good | 7 | \( 1 + 5 T + p T^{2} \) | 1.7.f |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 13 T + p T^{2} \) | 1.53.an |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−9.330690351644114393799101327461, −8.893742724875141354033404930208, −7.69597605358489864554393194674, −6.90040839368604728165720643072, −6.22998341966296941105214493465, −5.68225202127756533842843898175, −4.05959605040213207878208190727, −3.27500002241322509681556517668, −2.22266410857107931977491765784, −0.77331139176634990855171841029,
0.77331139176634990855171841029, 2.22266410857107931977491765784, 3.27500002241322509681556517668, 4.05959605040213207878208190727, 5.68225202127756533842843898175, 6.22998341966296941105214493465, 6.90040839368604728165720643072, 7.69597605358489864554393194674, 8.893742724875141354033404930208, 9.330690351644114393799101327461
