| L(s) = 1 | − 3-s + 5-s − 3·7-s − 2·9-s + 2·11-s − 15-s − 5·17-s − 2·19-s + 3·21-s − 4·23-s − 4·25-s + 5·27-s + 2·29-s + 8·31-s − 2·33-s − 3·35-s − 7·37-s − 6·41-s − 3·43-s − 2·45-s − 9·47-s + 2·49-s + 5·51-s + 10·53-s + 2·55-s + 2·57-s − 6·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.13·7-s − 2/3·9-s + 0.603·11-s − 0.258·15-s − 1.21·17-s − 0.458·19-s + 0.654·21-s − 0.834·23-s − 4/5·25-s + 0.962·27-s + 0.371·29-s + 1.43·31-s − 0.348·33-s − 0.507·35-s − 1.15·37-s − 0.937·41-s − 0.457·43-s − 0.298·45-s − 1.31·47-s + 2/7·49-s + 0.700·51-s + 1.37·53-s + 0.269·55-s + 0.264·57-s − 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 305552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 305552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| 113 | \( 1 - T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 3 T + p T^{2} \) | 1.43.d |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 11 T + p T^{2} \) | 1.71.l |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−12.98574649335703, −12.25177910048144, −11.95713639747309, −11.68336600462306, −11.11769577601418, −10.52090022755463, −10.06365556656971, −9.945726049724417, −9.114700449118682, −8.876756732764579, −8.391339519602905, −7.874887266253414, −7.053007823285057, −6.650187587431546, −6.256795857442910, −6.088198703328172, −5.421489089381776, −4.848943620841966, −4.322170434269072, −3.802120890793931, −3.095680884989014, −2.771755675284175, −1.974289005500141, −1.535472049667190, −0.4915373596385580, 0,
0.4915373596385580, 1.535472049667190, 1.974289005500141, 2.771755675284175, 3.095680884989014, 3.802120890793931, 4.322170434269072, 4.848943620841966, 5.421489089381776, 6.088198703328172, 6.256795857442910, 6.650187587431546, 7.053007823285057, 7.874887266253414, 8.391339519602905, 8.876756732764579, 9.114700449118682, 9.945726049724417, 10.06365556656971, 10.52090022755463, 11.11769577601418, 11.68336600462306, 11.95713639747309, 12.25177910048144, 12.98574649335703
