| L(s) = 1 | + 4·7-s + 13-s + 17-s − 19-s − 6·23-s − 5·25-s − 4·29-s − 4·31-s − 12·37-s + 2·41-s + 43-s + 47-s + 9·49-s + 6·53-s + 12·59-s + 6·61-s − 3·67-s − 4·71-s + 10·73-s + 4·79-s + 5·83-s + 13·89-s + 4·91-s − 6·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 0.277·13-s + 0.242·17-s − 0.229·19-s − 1.25·23-s − 25-s − 0.742·29-s − 0.718·31-s − 1.97·37-s + 0.312·41-s + 0.152·43-s + 0.145·47-s + 9/7·49-s + 0.824·53-s + 1.56·59-s + 0.768·61-s − 0.366·67-s − 0.474·71-s + 1.17·73-s + 0.450·79-s + 0.548·83-s + 1.37·89-s + 0.419·91-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 296208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 12 T + p T^{2} \) | 1.37.m |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 5 T + p T^{2} \) | 1.83.af |
| 89 | \( 1 - 13 T + p T^{2} \) | 1.89.an |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−12.99762390112716, −12.27066924708741, −11.93011205017917, −11.68335187090008, −11.02238700391127, −10.74347245223827, −10.28046744912940, −9.708593825374859, −9.255480930132117, −8.554949134859913, −8.383612340946375, −7.846082242024076, −7.417508154049021, −6.974141584005000, −6.304746128048023, −5.649477132309495, −5.434133857406890, −4.903081973003504, −4.235626482531274, −3.799903744129547, −3.454504508209714, −2.361800660212266, −2.026044826405260, −1.604249693279683, −0.8276884217441236, 0,
0.8276884217441236, 1.604249693279683, 2.026044826405260, 2.361800660212266, 3.454504508209714, 3.799903744129547, 4.235626482531274, 4.903081973003504, 5.434133857406890, 5.649477132309495, 6.304746128048023, 6.974141584005000, 7.417508154049021, 7.846082242024076, 8.383612340946375, 8.554949134859913, 9.255480930132117, 9.708593825374859, 10.28046744912940, 10.74347245223827, 11.02238700391127, 11.68335187090008, 11.93011205017917, 12.27066924708741, 12.99762390112716
