| L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 2·6-s + 9-s − 6·11-s + 2·12-s − 4·13-s − 4·16-s + 7·17-s − 2·18-s + 5·19-s + 12·22-s − 23-s − 5·25-s + 8·26-s + 27-s + 29-s + 4·31-s + 8·32-s − 6·33-s − 14·34-s + 2·36-s + 3·37-s − 10·38-s − 4·39-s − 6·43-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 0.816·6-s + 1/3·9-s − 1.80·11-s + 0.577·12-s − 1.10·13-s − 16-s + 1.69·17-s − 0.471·18-s + 1.14·19-s + 2.55·22-s − 0.208·23-s − 25-s + 1.56·26-s + 0.192·27-s + 0.185·29-s + 0.718·31-s + 1.41·32-s − 1.04·33-s − 2.40·34-s + 1/3·36-s + 0.493·37-s − 1.62·38-s − 0.640·39-s − 0.914·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 67 | \( 1 + T \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−7.57103193822429983060397848080, −7.24845967787942788762634754478, −6.08127957583156452565690451023, −5.20819975268529000155020887599, −4.74207132337061781550354641531, −3.47202294908723044695164261535, −2.76237152828383054677269248733, −2.07801678250387393248287011230, −1.05124651536894343232120694256, 0,
1.05124651536894343232120694256, 2.07801678250387393248287011230, 2.76237152828383054677269248733, 3.47202294908723044695164261535, 4.74207132337061781550354641531, 5.20819975268529000155020887599, 6.08127957583156452565690451023, 7.24845967787942788762634754478, 7.57103193822429983060397848080
