| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s − 4·11-s + 12-s + 6·13-s − 14-s + 16-s + 17-s − 18-s − 19-s + 21-s + 4·22-s − 24-s − 5·25-s − 6·26-s + 27-s + 28-s + 4·29-s − 7·31-s − 32-s − 4·33-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 − 1.20·11-s + 0.288·12-s + 1.66·13-s − 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.229·19-s + 0.218·21-s + 0.852·22-s − 0.204·24-s − 25-s − 1.17·26-s + 0.192·27-s + 0.188·28-s + 0.742·29-s − 1.25·31-s − 0.176·32-s − 0.696·33-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53958 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53958 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.667978614\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.667978614\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 17 | \( 1 - T \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 3 T + p T^{2} \) | 1.61.d |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−14.28755053634031, −13.99222164532735, −13.46566923882923, −12.92397775912967, −12.47846550086173, −11.74283710401253, −11.20649183500620, −10.77656446596719, −10.25298463270613, −9.887997704646046, −9.015101871382486, −8.742574547513278, −8.167840042481540, −7.816768323154624, −7.273381625250332, −6.537428196566893, −6.018251855422593, −5.350412017153661, −4.794302586509438, −3.836988144217541, −3.428035061680770, −2.758880714251354, −1.858343278617940, −1.560131577144963, −0.4736232089554398,
0.4736232089554398, 1.560131577144963, 1.858343278617940, 2.758880714251354, 3.428035061680770, 3.836988144217541, 4.794302586509438, 5.350412017153661, 6.018251855422593, 6.537428196566893, 7.273381625250332, 7.816768323154624, 8.167840042481540, 8.742574547513278, 9.015101871382486, 9.887997704646046, 10.25298463270613, 10.77656446596719, 11.20649183500620, 11.74283710401253, 12.47846550086173, 12.92397775912967, 13.46566923882923, 13.99222164532735, 14.28755053634031
