| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s + 3·11-s + 13-s + 15-s + 17-s − 6·19-s + 21-s + 6·23-s − 4·25-s − 27-s − 2·29-s − 4·31-s − 3·33-s + 35-s − 9·37-s − 39-s − 7·43-s − 45-s + 12·47-s + 49-s − 51-s + 13·53-s − 3·55-s + 6·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.904·11-s + 0.277·13-s + 0.258·15-s + 0.242·17-s − 1.37·19-s + 0.218·21-s + 1.25·23-s − 4/5·25-s − 0.192·27-s − 0.371·29-s − 0.718·31-s − 0.522·33-s + 0.169·35-s − 1.47·37-s − 0.160·39-s − 1.06·43-s − 0.149·45-s + 1.75·47-s + 1/7·49-s − 0.140·51-s + 1.78·53-s − 0.404·55-s + 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.112509367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.112509367\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 9 T + p T^{2} \) | 1.37.j |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 13 T + p T^{2} \) | 1.53.an |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 - T + p T^{2} \) | 1.83.ab |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−15.48667229591338, −14.95167187423055, −14.64984501319325, −13.60061911389090, −13.46391835269156, −12.62717100692548, −12.12910179091944, −11.84638083853110, −10.95041459179831, −10.78094081682521, −10.07992320353585, −9.335705372676097, −8.846206948529134, −8.363155717740219, −7.404398958291202, −7.030749275769652, −6.431363986714562, −5.815853732397044, −5.217633064860817, −4.377058248769697, −3.850091465581208, −3.309048717409457, −2.223652183289430, −1.437706594115997, −0.4547926342825364,
0.4547926342825364, 1.437706594115997, 2.223652183289430, 3.309048717409457, 3.850091465581208, 4.377058248769697, 5.217633064860817, 5.815853732397044, 6.431363986714562, 7.030749275769652, 7.404398958291202, 8.363155717740219, 8.846206948529134, 9.335705372676097, 10.07992320353585, 10.78094081682521, 10.95041459179831, 11.84638083853110, 12.12910179091944, 12.62717100692548, 13.46391835269156, 13.60061911389090, 14.64984501319325, 14.95167187423055, 15.48667229591338
