| L(s) = 1 | − 3-s + 5-s − 7-s + 9-s − 5·11-s + 5·13-s − 15-s + 17-s + 2·19-s + 21-s − 2·23-s − 4·25-s − 27-s − 2·29-s + 8·31-s + 5·33-s − 35-s − 7·37-s − 5·39-s − 12·41-s − 5·43-s + 45-s − 12·47-s + 49-s − 51-s + 53-s − 5·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.50·11-s + 1.38·13-s − 0.258·15-s + 0.242·17-s + 0.458·19-s + 0.218·21-s − 0.417·23-s − 4/5·25-s − 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.870·33-s − 0.169·35-s − 1.15·37-s − 0.800·39-s − 1.87·41-s − 0.762·43-s + 0.149·45-s − 1.75·47-s + 1/7·49-s − 0.140·51-s + 0.137·53-s − 0.674·55-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
| \(L(1)\) |
\(\approx\) |
\(1.255346528\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.255346528\) |
| \(L(\frac{3}{2})\) |
|
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.ab |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - T + p T^{2} \) | 1.67.ab |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + 5 T + p T^{2} \) | 1.83.f |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 - 15 T + p T^{2} \) | 1.97.ap |
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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
−15.61024504109878, −15.27123609368850, −14.21042384279062, −13.72330128478265, −13.30114081650902, −12.93842091251850, −12.22533380473948, −11.55779871395518, −11.22621603892192, −10.39678917006250, −10.01405920423401, −9.765020461400637, −8.610159600842982, −8.339729164678357, −7.666007191034016, −6.869293286984390, −6.340717010701054, −5.776374241004596, −5.218559134687711, −4.716878272850304, −3.594991815841070, −3.276833931459702, −2.215589683370354, −1.528653497937852, −0.4664849090553893,
0.4664849090553893, 1.528653497937852, 2.215589683370354, 3.276833931459702, 3.594991815841070, 4.716878272850304, 5.218559134687711, 5.776374241004596, 6.340717010701054, 6.869293286984390, 7.666007191034016, 8.339729164678357, 8.610159600842982, 9.765020461400637, 10.01405920423401, 10.39678917006250, 11.22621603892192, 11.55779871395518, 12.22533380473948, 12.93842091251850, 13.30114081650902, 13.72330128478265, 14.21042384279062, 15.27123609368850, 15.61024504109878
