| L(s) = 1 | − 5·7-s + 2·13-s − 8·19-s − 5·25-s + 7·31-s − 37-s + 13·43-s + 18·49-s − 61-s − 5·67-s − 7·73-s + 4·79-s − 10·91-s + 14·97-s + 13·103-s + 17·109-s + ⋯ |
| L(s) = 1 | − 1.88·7-s + 0.554·13-s − 1.83·19-s − 25-s + 1.25·31-s − 0.164·37-s + 1.98·43-s + 18/7·49-s − 0.128·61-s − 0.610·67-s − 0.819·73-s + 0.450·79-s − 1.04·91-s + 1.42·97-s + 1.28·103-s + 1.62·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.060810880\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.060810880\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 5 T + p T^{2} \) | 1.7.f |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 13 T + p T^{2} \) | 1.43.an |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−8.651035333995681317080070810460, −7.70203378606080007864041810394, −6.85869920998694804771072772744, −6.14264118212333135885925672066, −5.91910758825818844146409607562, −4.48185997599379011166802508586, −3.84056353467085125832194834909, −3.00558512009287535497097773694, −2.14407033478337402696046561651, −0.56552617859346549953557177563,
0.56552617859346549953557177563, 2.14407033478337402696046561651, 3.00558512009287535497097773694, 3.84056353467085125832194834909, 4.48185997599379011166802508586, 5.91910758825818844146409607562, 6.14264118212333135885925672066, 6.85869920998694804771072772744, 7.70203378606080007864041810394, 8.651035333995681317080070810460
