| L(s) = 1 | − 2-s − 3-s + 4-s − 2·5-s + 6-s − 2·7-s − 8-s + 9-s + 2·10-s − 4·11-s − 12-s + 2·14-s + 2·15-s + 16-s − 18-s − 2·19-s − 2·20-s + 2·21-s + 4·22-s − 4·23-s + 24-s − 25-s − 27-s − 2·28-s − 6·29-s − 2·30-s + 6·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 1.20·11-s − 0.288·12-s + 0.534·14-s + 0.516·15-s + 1/4·16-s − 0.235·18-s − 0.458·19-s − 0.447·20-s + 0.436·21-s + 0.852·22-s − 0.834·23-s + 0.204·24-s − 1/5·25-s − 0.192·27-s − 0.377·28-s − 1.11·29-s − 0.365·30-s + 1.07·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 293046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 293046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1148802526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1148802526\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−12.57791584115500, −12.10570761111870, −11.83257038634605, −11.26238008771126, −10.84327314625404, −10.36644477961000, −10.09038825063499, −9.538745300277508, −9.081984734325018, −8.406364278808666, −8.025038513053963, −7.698133909750771, −7.136853233195415, −6.703353545032351, −6.200964744720692, −5.606539054436246, −5.296105390335106, −4.501550326825264, −3.999120678524046, −3.541950962216440, −2.855707596052941, −2.341073300440730, −1.706958204073157, −0.8030800876029520, −0.1392619480097579,
0.1392619480097579, 0.8030800876029520, 1.706958204073157, 2.341073300440730, 2.855707596052941, 3.541950962216440, 3.999120678524046, 4.501550326825264, 5.296105390335106, 5.606539054436246, 6.200964744720692, 6.703353545032351, 7.136853233195415, 7.698133909750771, 8.025038513053963, 8.406364278808666, 9.081984734325018, 9.538745300277508, 10.09038825063499, 10.36644477961000, 10.84327314625404, 11.26238008771126, 11.83257038634605, 12.10570761111870, 12.57791584115500
