| L(s) = 1 | + 2-s − 3-s − 4-s − 6-s − 7-s − 3·8-s − 2·9-s + 12-s + 13-s − 14-s − 16-s − 7·17-s − 2·18-s − 7·19-s + 21-s + 23-s + 3·24-s − 5·25-s + 26-s + 5·27-s + 28-s + 9·29-s + 4·31-s + 5·32-s − 7·34-s + 2·36-s − 3·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 0.377·7-s − 1.06·8-s − 2/3·9-s + 0.288·12-s + 0.277·13-s − 0.267·14-s − 1/4·16-s − 1.69·17-s − 0.471·18-s − 1.60·19-s + 0.218·21-s + 0.208·23-s + 0.612·24-s − 25-s + 0.196·26-s + 0.962·27-s + 0.188·28-s + 1.67·29-s + 0.718·31-s + 0.883·32-s − 1.20·34-s + 1/3·36-s − 0.493·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 371 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 371 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | 7 | \( 1 + T \) | |
| 53 | \( 1 + T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + 11 T + p T^{2} \) | 1.83.l |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 11 T + p T^{2} \) | 1.97.l |
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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
−11.08477683809803031611129780994, −10.19113118748214175356034898084, −8.908232344565644566493262891414, −8.422868829470044189214539455019, −6.59016722970579549373610466360, −6.10026069334748150125541686180, −4.90399558150416474100355981272, −4.07348635622139479294973747045, −2.65706563163605891202165369262, 0,
2.65706563163605891202165369262, 4.07348635622139479294973747045, 4.90399558150416474100355981272, 6.10026069334748150125541686180, 6.59016722970579549373610466360, 8.422868829470044189214539455019, 8.908232344565644566493262891414, 10.19113118748214175356034898084, 11.08477683809803031611129780994
