| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 2·7-s + 8-s − 2·9-s − 4·11-s + 12-s − 4·13-s + 2·14-s + 16-s + 3·17-s − 2·18-s − 19-s + 2·21-s − 4·22-s + 24-s − 4·26-s − 5·27-s + 2·28-s + 8·29-s + 2·31-s + 32-s − 4·33-s + 3·34-s − 2·36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.755·7-s + 0.353·8-s − 2/3·9-s − 1.20·11-s + 0.288·12-s − 1.10·13-s + 0.534·14-s + 1/4·16-s + 0.727·17-s − 0.471·18-s − 0.229·19-s + 0.436·21-s − 0.852·22-s + 0.204·24-s − 0.784·26-s − 0.962·27-s + 0.377·28-s + 1.48·29-s + 0.359·31-s + 0.176·32-s − 0.696·33-s + 0.514·34-s − 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 3 T + p T^{2} \) | 1.43.ad |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 7 T + p T^{2} \) | 1.59.ah |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 7 T + p T^{2} \) | 1.83.ah |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−15.38857442171329, −14.74235216118968, −14.55269491605587, −14.10114858399317, −13.49134324697182, −12.99087096092590, −12.33488359455989, −11.95148595168903, −11.29999288536864, −10.81738131949758, −10.02938668591592, −9.841407088180856, −8.709789534354685, −8.416458160067346, −7.682624725319261, −7.471383357241144, −6.591131398314000, −5.775798548501247, −5.330060618903177, −4.753037086123403, −4.200981229735955, −3.206021962705519, −2.690354740480423, −2.290998287797164, −1.257466815720278, 0,
1.257466815720278, 2.290998287797164, 2.690354740480423, 3.206021962705519, 4.200981229735955, 4.753037086123403, 5.330060618903177, 5.775798548501247, 6.591131398314000, 7.471383357241144, 7.682624725319261, 8.416458160067346, 8.709789534354685, 9.841407088180856, 10.02938668591592, 10.81738131949758, 11.29999288536864, 11.95148595168903, 12.33488359455989, 12.99087096092590, 13.49134324697182, 14.10114858399317, 14.55269491605587, 14.74235216118968, 15.38857442171329
