| L(s) = 1 | + 2-s − 3-s + 4-s + 3·5-s − 6-s + 8-s + 9-s + 3·10-s − 11-s − 12-s + 4·13-s − 3·15-s + 16-s + 3·17-s + 18-s − 2·19-s + 3·20-s − 22-s − 24-s + 4·25-s + 4·26-s − 27-s + 6·29-s − 3·30-s − 2·31-s + 32-s + 33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.948·10-s − 0.301·11-s − 0.288·12-s + 1.10·13-s − 0.774·15-s + 1/4·16-s + 0.727·17-s + 0.235·18-s − 0.458·19-s + 0.670·20-s − 0.213·22-s − 0.204·24-s + 4/5·25-s + 0.784·26-s − 0.192·27-s + 1.11·29-s − 0.547·30-s − 0.359·31-s + 0.176·32-s + 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119658 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119658 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.534327413\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.534327413\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 37 | \( 1 - T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−13.64671378783899, −13.10856253729558, −12.66634937037077, −12.29061975610932, −11.63543154298929, −11.16694281304517, −10.66937452569070, −10.24329149955110, −9.877147676016351, −9.249017484724638, −8.699932300994397, −8.068309519118034, −7.578760095576121, −6.746744589050427, −6.346240566048426, −6.109439712748281, −5.445540838830760, −5.112987632476027, −4.543304503641710, −3.788713779264484, −3.269714451600963, −2.568749280375988, −1.912288321828250, −1.388631935650978, −0.6675061868426345,
0.6675061868426345, 1.388631935650978, 1.912288321828250, 2.568749280375988, 3.269714451600963, 3.788713779264484, 4.543304503641710, 5.112987632476027, 5.445540838830760, 6.109439712748281, 6.346240566048426, 6.746744589050427, 7.578760095576121, 8.068309519118034, 8.699932300994397, 9.249017484724638, 9.877147676016351, 10.24329149955110, 10.66937452569070, 11.16694281304517, 11.63543154298929, 12.29061975610932, 12.66634937037077, 13.10856253729558, 13.64671378783899
