| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 7-s + 8-s + 9-s + 6·11-s + 12-s − 4·13-s + 14-s + 16-s + 17-s + 18-s − 4·19-s + 21-s + 6·22-s + 8·23-s + 24-s − 4·26-s + 27-s + 28-s + 4·29-s + 8·31-s + 32-s + 6·33-s + 34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 1.80·11-s + 0.288·12-s − 1.10·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.917·19-s + 0.218·21-s + 1.27·22-s + 1.66·23-s + 0.204·24-s − 0.784·26-s + 0.192·27-s + 0.188·28-s + 0.742·29-s + 1.43·31-s + 0.176·32-s + 1.04·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.865409097\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.865409097\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 - T \) | |
| good | 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−15.45451195162296, −15.11476128614319, −14.59679074530264, −14.34402565138866, −13.72985673428629, −13.08599849481400, −12.55023323391579, −11.96273641910697, −11.56344410054712, −10.94098258623393, −10.12922237340635, −9.676490072862720, −8.944167721160688, −8.499807383266336, −7.751464589534030, −7.040461599314517, −6.616145029387711, −6.038434496654913, −4.981295927504541, −4.569514719465389, −4.023291899893852, −3.136987462366937, −2.607258822281062, −1.683301994400179, −0.9460154769990073,
0.9460154769990073, 1.683301994400179, 2.607258822281062, 3.136987462366937, 4.023291899893852, 4.569514719465389, 4.981295927504541, 6.038434496654913, 6.616145029387711, 7.040461599314517, 7.751464589534030, 8.499807383266336, 8.944167721160688, 9.676490072862720, 10.12922237340635, 10.94098258623393, 11.56344410054712, 11.96273641910697, 12.55023323391579, 13.08599849481400, 13.72985673428629, 14.34402565138866, 14.59679074530264, 15.11476128614319, 15.45451195162296
