| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s + 9-s − 10-s − 11-s + 12-s − 4·13-s − 15-s + 16-s − 8·17-s + 18-s + 6·19-s − 20-s − 22-s + 4·23-s + 24-s + 25-s − 4·26-s + 27-s + 2·29-s − 30-s + 32-s − 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 − 0.301·11-s + 0.288·12-s − 1.10·13-s − 0.258·15-s + 1/4·16-s − 1.94·17-s + 0.235·18-s + 1.37·19-s − 0.223·20-s − 0.213·22-s + 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.784·26-s + 0.192·27-s + 0.371·29-s − 0.182·30-s + 0.176·32-s − 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 317130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.215323503\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.215323503\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 31 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 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.ac |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−12.71286075087317, −12.29733239856969, −11.73057712509791, −11.36710158120366, −10.89092566313574, −10.51161734945129, −9.825529546585484, −9.346258494154923, −9.094308121281845, −8.427327656014403, −7.872860177663876, −7.507068471831550, −7.080849419272109, −6.649935002852135, −6.098944032482591, −5.409727635598801, −4.856833571102004, −4.627632918377963, −4.041649871475037, −3.512411604657764, −2.752839028677400, −2.632778506210453, −2.015119513851518, −1.171444169360268, −0.4667239371399511,
0.4667239371399511, 1.171444169360268, 2.015119513851518, 2.632778506210453, 2.752839028677400, 3.512411604657764, 4.041649871475037, 4.627632918377963, 4.856833571102004, 5.409727635598801, 6.098944032482591, 6.649935002852135, 7.080849419272109, 7.507068471831550, 7.872860177663876, 8.427327656014403, 9.094308121281845, 9.346258494154923, 9.825529546585484, 10.51161734945129, 10.89092566313574, 11.36710158120366, 11.73057712509791, 12.29733239856969, 12.71286075087317
