| L(s) = 1 | − 2-s − 2·3-s − 4-s − 5-s + 2·6-s + 3·8-s + 9-s + 10-s + 2·12-s + 13-s + 2·15-s − 16-s − 5·17-s − 18-s + 6·19-s + 20-s + 2·23-s − 6·24-s − 4·25-s − 26-s + 4·27-s − 9·29-s − 2·30-s + 2·31-s − 5·32-s + 5·34-s − 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s − 0.447·5-s + 0.816·6-s + 1.06·8-s + 1/3·9-s + 0.316·10-s + 0.577·12-s + 0.277·13-s + 0.516·15-s − 1/4·16-s − 1.21·17-s − 0.235·18-s + 1.37·19-s + 0.223·20-s + 0.417·23-s − 1.22·24-s − 4/5·25-s − 0.196·26-s + 0.769·27-s − 1.67·29-s − 0.365·30-s + 0.359·31-s − 0.883·32-s + 0.857·34-s − 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | 7 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 - 13 T + p T^{2} \) | 1.97.an |
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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
−7.74876323699072704778810695090, −7.11362745110257617710521797400, −6.38296664456429484707652424372, −5.43069111134688314478692077104, −5.05042499311079542820546840649, −4.15190959135177339748970024081, −3.41971926023207922940384602095, −1.97805102262059112135043101795, −0.860092589317136083123344363004, 0,
0.860092589317136083123344363004, 1.97805102262059112135043101795, 3.41971926023207922940384602095, 4.15190959135177339748970024081, 5.05042499311079542820546840649, 5.43069111134688314478692077104, 6.38296664456429484707652424372, 7.11362745110257617710521797400, 7.74876323699072704778810695090
