| L(s) = 1 | − 2-s − 4-s + 3·8-s + 2·11-s + 4·13-s − 16-s − 4·17-s + 19-s − 2·22-s − 2·23-s − 5·25-s − 4·26-s + 2·29-s − 5·32-s + 4·34-s + 6·37-s − 38-s − 6·41-s + 8·43-s − 2·44-s + 2·46-s + 5·50-s − 4·52-s + 2·53-s − 2·58-s + 4·59-s + 10·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s + 0.603·11-s + 1.10·13-s − 1/4·16-s − 0.970·17-s + 0.229·19-s − 0.426·22-s − 0.417·23-s − 25-s − 0.784·26-s + 0.371·29-s − 0.883·32-s + 0.685·34-s + 0.986·37-s − 0.162·38-s − 0.937·41-s + 1.21·43-s − 0.301·44-s + 0.294·46-s + 0.707·50-s − 0.554·52-s + 0.274·53-s − 0.262·58-s + 0.520·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.160350318\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.160350318\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 19 | \( 1 - T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−8.074441240986641016381142623006, −7.19241917477318062285982057801, −6.51553200085163157183564465784, −5.80156872936238102919155140790, −4.98936980933135032801497052950, −4.02223010224911046316002801924, −3.82462885317567531324638393085, −2.45329369087640475373386960474, −1.51688116571328033763886442228, −0.63351366554247137292354589067,
0.63351366554247137292354589067, 1.51688116571328033763886442228, 2.45329369087640475373386960474, 3.82462885317567531324638393085, 4.02223010224911046316002801924, 4.98936980933135032801497052950, 5.80156872936238102919155140790, 6.51553200085163157183564465784, 7.19241917477318062285982057801, 8.074441240986641016381142623006
