L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 8-s + 9-s + 10-s − 4·11-s + 12-s + 2·13-s + 15-s + 16-s − 2·17-s + 18-s + 4·19-s + 20-s − 4·22-s + 8·23-s + 24-s + 25-s + 2·26-s + 27-s + 2·29-s + 30-s + 32-s − 4·33-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s + 0.288·12-s + 0.554·13-s + 0.258·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.852·22-s + 1.66·23-s + 0.204·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.371·29-s + 0.182·30-s + 0.176·32-s − 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.202823613\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.202823613\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 41 | \( 1 \) | |
good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 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.ag |
| 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
−14.39818431859722, −13.98054462173637, −13.38016607270447, −13.18003916607475, −12.66310711783701, −12.15787952986311, −11.28816801012904, −10.98001510045882, −10.48267529005919, −9.893754366737923, −9.133599239680479, −8.962356811009419, −8.065532590131538, −7.574000176418418, −7.205263557534746, −6.346721754267433, −5.974246673126924, −5.188450691725322, −4.837553908325307, −4.164332024078123, −3.353288230042658, −2.808062732414902, −2.449707150896666, −1.503791227147112, −0.7674036105234862,
0.7674036105234862, 1.503791227147112, 2.449707150896666, 2.808062732414902, 3.353288230042658, 4.164332024078123, 4.837553908325307, 5.188450691725322, 5.974246673126924, 6.346721754267433, 7.205263557534746, 7.574000176418418, 8.065532590131538, 8.962356811009419, 9.133599239680479, 9.893754366737923, 10.48267529005919, 10.98001510045882, 11.28816801012904, 12.15787952986311, 12.66310711783701, 13.18003916607475, 13.38016607270447, 13.98054462173637, 14.39818431859722