| L(s) = 1 | − 2-s + 4-s − 8-s + 6·11-s − 13-s + 16-s − 4·17-s − 2·19-s − 6·22-s + 6·23-s + 26-s + 10·29-s − 4·31-s − 32-s + 4·34-s − 6·37-s + 2·38-s + 10·41-s + 6·44-s − 6·46-s + 8·47-s − 52-s + 6·53-s − 10·58-s − 6·59-s + 6·61-s + 4·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 1.80·11-s − 0.277·13-s + 1/4·16-s − 0.970·17-s − 0.458·19-s − 1.27·22-s + 1.25·23-s + 0.196·26-s + 1.85·29-s − 0.718·31-s − 0.176·32-s + 0.685·34-s − 0.986·37-s + 0.324·38-s + 1.56·41-s + 0.904·44-s − 0.884·46-s + 1.16·47-s − 0.138·52-s + 0.824·53-s − 1.31·58-s − 0.781·59-s + 0.768·61-s + 0.508·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.154897892\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.154897892\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 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.e |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−12.57485482181476, −12.20731138383833, −11.73958945322853, −11.34192608014628, −10.82801334653685, −10.47305676776326, −9.961177499230726, −9.313716348483101, −8.939206845175343, −8.807100038833843, −8.287141037767541, −7.355470324780291, −7.259825505243764, −6.645489574716534, −6.305372219951952, −5.823400156553274, −5.058980695055436, −4.486140714640193, −4.097583011463665, −3.491516924835917, −2.787097319188086, −2.347452473895454, −1.591216840534956, −1.102626915994228, −0.4903421199994833,
0.4903421199994833, 1.102626915994228, 1.591216840534956, 2.347452473895454, 2.787097319188086, 3.491516924835917, 4.097583011463665, 4.486140714640193, 5.058980695055436, 5.823400156553274, 6.305372219951952, 6.645489574716534, 7.259825505243764, 7.355470324780291, 8.287141037767541, 8.807100038833843, 8.939206845175343, 9.313716348483101, 9.961177499230726, 10.47305676776326, 10.82801334653685, 11.34192608014628, 11.73958945322853, 12.20731138383833, 12.57485482181476
