| L(s) = 1 | − 2-s − 4-s + 4·7-s + 3·8-s + 4·11-s − 2·13-s − 4·14-s − 16-s − 6·17-s − 4·19-s − 4·22-s + 2·26-s − 4·28-s + 6·29-s − 31-s − 5·32-s + 6·34-s − 10·37-s + 4·38-s + 6·41-s + 12·43-s − 4·44-s + 9·49-s + 2·52-s − 2·53-s + 12·56-s − 6·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.51·7-s + 1.06·8-s + 1.20·11-s − 0.554·13-s − 1.06·14-s − 1/4·16-s − 1.45·17-s − 0.917·19-s − 0.852·22-s + 0.392·26-s − 0.755·28-s + 1.11·29-s − 0.179·31-s − 0.883·32-s + 1.02·34-s − 1.64·37-s + 0.648·38-s + 0.937·41-s + 1.82·43-s − 0.603·44-s + 9/7·49-s + 0.277·52-s − 0.274·53-s + 1.60·56-s − 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.365726562\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.365726562\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 31 | \( 1 + T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−8.203734725963423017657316921193, −7.32989529716010713511606044338, −6.81954214309764801604594478226, −5.84772949982742967064121289316, −4.84526032646291788774424960525, −4.47185750304142306057299379735, −3.86638259371708760268290533155, −2.33217926404402924601981347207, −1.68009312044061770417888485059, −0.69913093567049216504900024437,
0.69913093567049216504900024437, 1.68009312044061770417888485059, 2.33217926404402924601981347207, 3.86638259371708760268290533155, 4.47185750304142306057299379735, 4.84526032646291788774424960525, 5.84772949982742967064121289316, 6.81954214309764801604594478226, 7.32989529716010713511606044338, 8.203734725963423017657316921193
