| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 5·7-s − 8-s + 9-s − 3·11-s − 12-s − 4·13-s + 5·14-s + 16-s + 2·17-s − 18-s − 6·19-s + 5·21-s + 3·22-s − 6·23-s + 24-s + 4·26-s − 27-s − 5·28-s + 6·29-s − 32-s + 3·33-s − 2·34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.88·7-s − 0.353·8-s + 1/3·9-s − 0.904·11-s − 0.288·12-s − 1.10·13-s + 1.33·14-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 1.37·19-s + 1.09·21-s + 0.639·22-s − 1.25·23-s + 0.204·24-s + 0.784·26-s − 0.192·27-s − 0.944·28-s + 1.11·29-s − 0.176·32-s + 0.522·33-s − 0.342·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 277350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5078534794\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5078534794\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 43 | \( 1 \) | |
| good | 7 | \( 1 + 5 T + p T^{2} \) | 1.7.f |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 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.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 11 T + p T^{2} \) | 1.53.l |
| 59 | \( 1 - T + p T^{2} \) | 1.59.ab |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 15 T + p T^{2} \) | 1.67.p |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−12.52621864730952, −12.34885104088011, −12.01305420891504, −11.29296101834917, −10.69463991597650, −10.34397752601330, −10.04224050943125, −9.607870936848323, −9.298185995241946, −8.556522750963232, −8.104761744246730, −7.559746722206584, −7.180582902336998, −6.526536037031105, −6.152741756433145, −6.012715364464318, −5.138913271635605, −4.669456443663091, −4.074316791916222, −3.223125897817668, −3.058271058435594, −2.210129515823150, −1.929033299510178, −0.6561489333158416, −0.3439023393613809,
0.3439023393613809, 0.6561489333158416, 1.929033299510178, 2.210129515823150, 3.058271058435594, 3.223125897817668, 4.074316791916222, 4.669456443663091, 5.138913271635605, 6.012715364464318, 6.152741756433145, 6.526536037031105, 7.180582902336998, 7.559746722206584, 8.104761744246730, 8.556522750963232, 9.298185995241946, 9.607870936848323, 10.04224050943125, 10.34397752601330, 10.69463991597650, 11.29296101834917, 12.01305420891504, 12.34885104088011, 12.52621864730952
