| L(s) = 1 | − 3-s − 5-s + 5·7-s + 9-s + 3·11-s + 15-s − 8·17-s + 5·19-s − 5·21-s + 4·23-s + 25-s − 27-s − 4·29-s + 2·31-s − 3·33-s − 5·35-s − 7·37-s + 6·41-s − 6·43-s − 45-s + 3·47-s + 18·49-s + 8·51-s + 53-s − 3·55-s − 5·57-s − 12·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.88·7-s + 1/3·9-s + 0.904·11-s + 0.258·15-s − 1.94·17-s + 1.14·19-s − 1.09·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 0.742·29-s + 0.359·31-s − 0.522·33-s − 0.845·35-s − 1.15·37-s + 0.937·41-s − 0.914·43-s − 0.149·45-s + 0.437·47-s + 18/7·49-s + 1.12·51-s + 0.137·53-s − 0.404·55-s − 0.662·57-s − 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.264768811\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.264768811\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + 11 T + p T^{2} \) | 1.89.l |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−14.90415090556338, −14.14570959767815, −13.88884580879317, −13.27048040937971, −12.53827319126520, −11.98412225677989, −11.49584349128654, −11.22181960098213, −10.87312331013302, −10.20235605977252, −9.304632570303814, −8.885434321145141, −8.453918693309293, −7.711196522077003, −7.205591444056240, −6.823198871318719, −5.978825637426080, −5.379551296835601, −4.674686371255166, −4.498270503499701, −3.781659923709936, −2.878175153008610, −1.882837775694213, −1.468439798510843, −0.5947481620427121,
0.5947481620427121, 1.468439798510843, 1.882837775694213, 2.878175153008610, 3.781659923709936, 4.498270503499701, 4.674686371255166, 5.379551296835601, 5.978825637426080, 6.823198871318719, 7.205591444056240, 7.711196522077003, 8.453918693309293, 8.885434321145141, 9.304632570303814, 10.20235605977252, 10.87312331013302, 11.22181960098213, 11.49584349128654, 11.98412225677989, 12.53827319126520, 13.27048040937971, 13.88884580879317, 14.14570959767815, 14.90415090556338
