| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·6-s − 4·7-s + 8-s + 9-s + 11-s − 2·12-s − 2·13-s − 4·14-s + 16-s + 2·17-s + 18-s + 8·21-s + 22-s − 2·24-s − 2·26-s + 4·27-s − 4·28-s − 6·29-s + 6·31-s + 32-s − 2·33-s + 2·34-s + 36-s − 4·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.577·12-s − 0.554·13-s − 1.06·14-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 1.74·21-s + 0.213·22-s − 0.408·24-s − 0.392·26-s + 0.769·27-s − 0.755·28-s − 1.11·29-s + 1.07·31-s + 0.176·32-s − 0.348·33-s + 0.342·34-s + 1/6·36-s − 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 19 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 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 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−13.17591083288705, −12.70177565579337, −12.41924251492185, −11.97678190743563, −11.51767131373069, −11.15023561357720, −10.51875914554863, −10.02085229822109, −9.826114326808558, −9.134967048826752, −8.655539727944916, −7.854488533190101, −7.318551618812821, −6.870811012883364, −6.384626923977778, −6.025237960307617, −5.626934749783425, −5.071666779991039, −4.553628228269660, −3.950617378318653, −3.357411639337527, −2.925393737929996, −2.302898432886576, −1.427744405816366, −0.6508241755256568, 0,
0.6508241755256568, 1.427744405816366, 2.302898432886576, 2.925393737929996, 3.357411639337527, 3.950617378318653, 4.553628228269660, 5.071666779991039, 5.626934749783425, 6.025237960307617, 6.384626923977778, 6.870811012883364, 7.318551618812821, 7.854488533190101, 8.655539727944916, 9.134967048826752, 9.826114326808558, 10.02085229822109, 10.51875914554863, 11.15023561357720, 11.51767131373069, 11.97678190743563, 12.41924251492185, 12.70177565579337, 13.17591083288705
