| L(s) = 1 | + 3-s − 4·7-s + 9-s + 6·11-s − 4·13-s − 4·17-s − 8·19-s − 4·21-s + 27-s − 2·29-s + 2·31-s + 6·33-s + 4·37-s − 4·39-s + 6·41-s + 12·43-s + 9·49-s − 4·51-s − 14·53-s − 8·57-s − 6·59-s + 6·61-s − 4·63-s + 4·67-s + 8·71-s + 2·73-s − 24·77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.80·11-s − 1.10·13-s − 0.970·17-s − 1.83·19-s − 0.872·21-s + 0.192·27-s − 0.371·29-s + 0.359·31-s + 1.04·33-s + 0.657·37-s − 0.640·39-s + 0.937·41-s + 1.82·43-s + 9/7·49-s − 0.560·51-s − 1.92·53-s − 1.05·57-s − 0.781·59-s + 0.768·61-s − 0.503·63-s + 0.488·67-s + 0.949·71-s + 0.234·73-s − 2.73·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.682270909\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.682270909\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−7.56267091631938505701742416011, −6.91990739390325624262535768722, −6.35135435314637763431378177121, −6.02932942518266175062379417914, −4.55269338970721426288778111897, −4.20591480053327202183564817207, −3.44776460462391349560872957344, −2.59058021220271205438755781686, −1.94671898744083558690806732935, −0.57501942029248595301015827103,
0.57501942029248595301015827103, 1.94671898744083558690806732935, 2.59058021220271205438755781686, 3.44776460462391349560872957344, 4.20591480053327202183564817207, 4.55269338970721426288778111897, 6.02932942518266175062379417914, 6.35135435314637763431378177121, 6.91990739390325624262535768722, 7.56267091631938505701742416011
