| L(s) = 1 | + 2·2-s + 2·4-s − 3·5-s − 6·10-s − 11-s − 2·13-s − 4·16-s − 17-s + 19-s − 6·20-s − 2·22-s + 4·23-s + 4·25-s − 4·26-s + 2·29-s + 6·31-s − 8·32-s − 2·34-s + 2·38-s − 43-s − 2·44-s + 8·46-s − 9·47-s + 8·50-s − 4·52-s − 10·53-s + 3·55-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 1.34·5-s − 1.89·10-s − 0.301·11-s − 0.554·13-s − 16-s − 0.242·17-s + 0.229·19-s − 1.34·20-s − 0.426·22-s + 0.834·23-s + 4/5·25-s − 0.784·26-s + 0.371·29-s + 1.07·31-s − 1.41·32-s − 0.342·34-s + 0.324·38-s − 0.152·43-s − 0.301·44-s + 1.17·46-s − 1.31·47-s + 1.13·50-s − 0.554·52-s − 1.37·53-s + 0.404·55-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
| \(L(1)\) |
\(\approx\) |
\(2.424620951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.424620951\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−7.85582869646283906731190811336, −6.77124936049535641641720020484, −6.54381789640621822379739731453, −5.42089478131328865490166156321, −4.80004704636396746296312994207, −4.43252527426558787350954315583, −3.48189187269012783645528672334, −3.11587306879081144724928834922, −2.15535598087150874841625030639, −0.58876266921858160851854225947,
0.58876266921858160851854225947, 2.15535598087150874841625030639, 3.11587306879081144724928834922, 3.48189187269012783645528672334, 4.43252527426558787350954315583, 4.80004704636396746296312994207, 5.42089478131328865490166156321, 6.54381789640621822379739731453, 6.77124936049535641641720020484, 7.85582869646283906731190811336
