| L(s) = 1 | − 3-s − 3·7-s − 2·9-s − 2·11-s + 13-s − 5·17-s − 19-s + 3·21-s + 23-s − 5·25-s + 5·27-s − 3·29-s − 4·31-s + 2·33-s + 2·37-s − 39-s − 8·41-s + 8·43-s + 8·47-s + 2·49-s + 5·51-s + 9·53-s + 57-s − 59-s + 14·61-s + 6·63-s − 13·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.13·7-s − 2/3·9-s − 0.603·11-s + 0.277·13-s − 1.21·17-s − 0.229·19-s + 0.654·21-s + 0.208·23-s − 25-s + 0.962·27-s − 0.557·29-s − 0.718·31-s + 0.348·33-s + 0.328·37-s − 0.160·39-s − 1.24·41-s + 1.21·43-s + 1.16·47-s + 2/7·49-s + 0.700·51-s + 1.23·53-s + 0.132·57-s − 0.130·59-s + 1.79·61-s + 0.755·63-s − 1.58·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + T + p T^{2} \) | 1.59.b |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 10 T + p T^{2} \) | 1.83.k |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 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
−11.20072332430483839043066410359, −10.47040915981345192829319556956, −9.389846141086456194104312076880, −8.517458100677749789097011376696, −7.18297506166153508271072468025, −6.21150502900434643631796858377, −5.41676012615338774687665516924, −3.93346432196685798564362812764, −2.56005012019129152265351485884, 0,
2.56005012019129152265351485884, 3.93346432196685798564362812764, 5.41676012615338774687665516924, 6.21150502900434643631796858377, 7.18297506166153508271072468025, 8.517458100677749789097011376696, 9.389846141086456194104312076880, 10.47040915981345192829319556956, 11.20072332430483839043066410359
