| L(s) = 1 | − 3-s − 5-s + 4·7-s + 9-s + 13-s + 15-s + 2·17-s + 4·19-s − 4·21-s − 8·23-s + 25-s − 27-s − 2·29-s + 8·31-s − 4·35-s + 2·37-s − 39-s + 6·41-s + 12·43-s − 45-s + 9·49-s − 2·51-s + 10·53-s − 4·57-s + 10·61-s + 4·63-s − 65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s + 0.277·13-s + 0.258·15-s + 0.485·17-s + 0.917·19-s − 0.872·21-s − 1.66·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.676·35-s + 0.328·37-s − 0.160·39-s + 0.937·41-s + 1.82·43-s − 0.149·45-s + 9/7·49-s − 0.280·51-s + 1.37·53-s − 0.529·57-s + 1.28·61-s + 0.503·63-s − 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 377520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 377520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.161651943\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.161651943\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 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.o |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−12.31144746698252, −11.88761128088741, −11.67196505112919, −11.13665552024459, −10.85726849776645, −10.33870952419406, −9.725076373214865, −9.508453059424375, −8.682379194816095, −8.251319726816165, −7.856179576967524, −7.648461392708184, −6.964134870954364, −6.520662864191357, −5.697053464312223, −5.598552853597938, −5.076428090261258, −4.386488871682115, −4.056450249429859, −3.725761327843047, −2.651110755534868, −2.394797596525965, −1.522840278194372, −1.062450711566911, −0.5507145275202251,
0.5507145275202251, 1.062450711566911, 1.522840278194372, 2.394797596525965, 2.651110755534868, 3.725761327843047, 4.056450249429859, 4.386488871682115, 5.076428090261258, 5.598552853597938, 5.697053464312223, 6.520662864191357, 6.964134870954364, 7.648461392708184, 7.856179576967524, 8.251319726816165, 8.682379194816095, 9.508453059424375, 9.725076373214865, 10.33870952419406, 10.85726849776645, 11.13665552024459, 11.67196505112919, 11.88761128088741, 12.31144746698252
