| L(s) = 1 | − 2-s + 4-s − 8-s − 3·9-s + 4·11-s + 3·13-s + 16-s + 2·17-s + 3·18-s − 6·19-s − 4·22-s − 6·23-s − 3·26-s − 6·29-s − 11·31-s − 32-s − 2·34-s − 3·36-s + 4·37-s + 6·38-s − 8·41-s − 10·43-s + 4·44-s + 6·46-s − 7·47-s + 3·52-s + 2·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 9-s + 1.20·11-s + 0.832·13-s + 1/4·16-s + 0.485·17-s + 0.707·18-s − 1.37·19-s − 0.852·22-s − 1.25·23-s − 0.588·26-s − 1.11·29-s − 1.97·31-s − 0.176·32-s − 0.342·34-s − 1/2·36-s + 0.657·37-s + 0.973·38-s − 1.24·41-s − 1.52·43-s + 0.603·44-s + 0.884·46-s − 1.02·47-s + 0.416·52-s + 0.274·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.04047379525\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04047379525\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 79 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 11 T + p T^{2} \) | 1.31.l |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + T + p T^{2} \) | 1.59.b |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 5 T + p T^{2} \) | 1.89.af |
| 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
−13.22042997708541, −12.40757316496773, −12.10732771263430, −11.53312393474938, −11.20935041286514, −10.83340154208937, −10.22193212254117, −9.758330142811636, −9.176349668471668, −8.824680568048891, −8.454754861965494, −7.942666519135144, −7.474968439701541, −6.677568200603629, −6.458093855774642, −5.828796081490340, −5.579986528066229, −4.715330994064002, −4.018449502616807, −3.524494289343505, −3.202811634510563, −2.171687501666166, −1.784035439826443, −1.270438403116587, −0.06280088351495491,
0.06280088351495491, 1.270438403116587, 1.784035439826443, 2.171687501666166, 3.202811634510563, 3.524494289343505, 4.018449502616807, 4.715330994064002, 5.579986528066229, 5.828796081490340, 6.458093855774642, 6.677568200603629, 7.474968439701541, 7.942666519135144, 8.454754861965494, 8.824680568048891, 9.176349668471668, 9.758330142811636, 10.22193212254117, 10.83340154208937, 11.20935041286514, 11.53312393474938, 12.10732771263430, 12.40757316496773, 13.22042997708541
