| L(s) = 1 | − 2-s + 4-s + 5-s + 5·7-s − 8-s − 10-s − 11-s − 5·14-s + 16-s + 4·17-s + 20-s + 22-s − 4·23-s + 25-s + 5·28-s + 4·29-s + 5·31-s − 32-s − 4·34-s + 5·35-s + 2·37-s − 40-s + 4·43-s − 44-s + 4·46-s + 47-s + 18·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 1.88·7-s − 0.353·8-s − 0.316·10-s − 0.301·11-s − 1.33·14-s + 1/4·16-s + 0.970·17-s + 0.223·20-s + 0.213·22-s − 0.834·23-s + 1/5·25-s + 0.944·28-s + 0.742·29-s + 0.898·31-s − 0.176·32-s − 0.685·34-s + 0.845·35-s + 0.328·37-s − 0.158·40-s + 0.609·43-s − 0.150·44-s + 0.589·46-s + 0.145·47-s + 18/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.702211333\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.702211333\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 263 | \( 1 - T \) | |
| good | 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 - 11 T + p T^{2} \) | 1.53.al |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 9 T + p T^{2} \) | 1.61.aj |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 13 T + p T^{2} \) | 1.73.an |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−15.60967095611319, −14.75283968290908, −14.46808731572353, −13.96781272401453, −13.44046243813806, −12.56848426495078, −11.94006799577895, −11.69310286432848, −10.95443831849308, −10.47339693568202, −10.05482875681424, −9.368061063946342, −8.675520171408272, −8.116287805906313, −7.874174577674217, −7.209991372353862, −6.437902905948014, −5.679688799210056, −5.250370364757405, −4.542358810085830, −3.852808471317145, −2.735019218887329, −2.195116100427802, −1.406585060245720, −0.8085025311357661,
0.8085025311357661, 1.406585060245720, 2.195116100427802, 2.735019218887329, 3.852808471317145, 4.542358810085830, 5.250370364757405, 5.679688799210056, 6.437902905948014, 7.209991372353862, 7.874174577674217, 8.116287805906313, 8.675520171408272, 9.368061063946342, 10.05482875681424, 10.47339693568202, 10.95443831849308, 11.69310286432848, 11.94006799577895, 12.56848426495078, 13.44046243813806, 13.96781272401453, 14.46808731572353, 14.75283968290908, 15.60967095611319
