| L(s) = 1 | − 3-s + 7-s + 9-s − 3·11-s − 13-s + 7·17-s − 4·19-s − 21-s − 9·23-s − 27-s + 8·29-s + 4·31-s + 3·33-s + 3·37-s + 39-s − 5·41-s − 2·47-s − 6·49-s − 7·51-s + 3·53-s + 4·57-s − 12·59-s − 15·61-s + 63-s + 12·67-s + 9·69-s − 15·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.904·11-s − 0.277·13-s + 1.69·17-s − 0.917·19-s − 0.218·21-s − 1.87·23-s − 0.192·27-s + 1.48·29-s + 0.718·31-s + 0.522·33-s + 0.493·37-s + 0.160·39-s − 0.780·41-s − 0.291·47-s − 6/7·49-s − 0.980·51-s + 0.412·53-s + 0.529·57-s − 1.56·59-s − 1.92·61-s + 0.125·63-s + 1.46·67-s + 1.08·69-s − 1.78·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 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 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 15 T + p T^{2} \) | 1.61.p |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 - 5 T + p T^{2} \) | 1.89.af |
| 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
−7.88034636577311951967712508010, −7.65780049200228864419114718114, −6.38060768440323609490686189736, −5.98987978926658342477866795777, −5.02404371431265524593916271340, −4.52314779190144249482201689447, −3.43190624197718900891926800981, −2.44246323218658512812099286972, −1.35266374108685457664484912560, 0,
1.35266374108685457664484912560, 2.44246323218658512812099286972, 3.43190624197718900891926800981, 4.52314779190144249482201689447, 5.02404371431265524593916271340, 5.98987978926658342477866795777, 6.38060768440323609490686189736, 7.65780049200228864419114718114, 7.88034636577311951967712508010
