| L(s) = 1 | − 2-s + 4-s − 3·7-s − 8-s + 11-s + 3·14-s + 16-s + 5·19-s − 22-s − 4·23-s − 3·28-s − 10·31-s − 32-s − 37-s − 5·38-s + 6·41-s + 2·43-s + 44-s + 4·46-s + 9·47-s + 2·49-s − 13·53-s + 3·56-s + 4·59-s − 2·61-s + 10·62-s + 64-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.13·7-s − 0.353·8-s + 0.301·11-s + 0.801·14-s + 1/4·16-s + 1.14·19-s − 0.213·22-s − 0.834·23-s − 0.566·28-s − 1.79·31-s − 0.176·32-s − 0.164·37-s − 0.811·38-s + 0.937·41-s + 0.304·43-s + 0.150·44-s + 0.589·46-s + 1.31·47-s + 2/7·49-s − 1.78·53-s + 0.400·56-s + 0.520·59-s − 0.256·61-s + 1.27·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6734564253\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6734564253\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 + 13 T + p T^{2} \) | 1.53.n |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - T + p T^{2} \) | 1.89.ab |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−14.20648081754158, −13.51511810004105, −12.91248981203961, −12.57382156192444, −12.04635877208117, −11.45060643668573, −11.08571634744128, −10.30894519218174, −10.03789279040159, −9.467057598474680, −9.018119670391663, −8.721465174759848, −7.746605161156943, −7.434199759881052, −7.042031831966866, −6.184704040056851, −5.956358123885282, −5.366726905242313, −4.454871512095661, −3.840681531536503, −3.210346527753940, −2.773980852279128, −1.889134856197872, −1.255082095827741, −0.3130426441260679,
0.3130426441260679, 1.255082095827741, 1.889134856197872, 2.773980852279128, 3.210346527753940, 3.840681531536503, 4.454871512095661, 5.366726905242313, 5.956358123885282, 6.184704040056851, 7.042031831966866, 7.434199759881052, 7.746605161156943, 8.721465174759848, 9.018119670391663, 9.467057598474680, 10.03789279040159, 10.30894519218174, 11.08571634744128, 11.45060643668573, 12.04635877208117, 12.57382156192444, 12.91248981203961, 13.51511810004105, 14.20648081754158
