| L(s) = 1 | + 2·3-s + 7-s + 9-s − 5·11-s − 7·13-s + 7·19-s + 2·21-s − 23-s − 4·27-s + 5·29-s + 10·31-s − 10·33-s − 2·37-s − 14·39-s + 3·41-s + 9·43-s + 8·47-s − 6·49-s − 4·53-s + 14·57-s + 2·59-s − 6·61-s + 63-s + 8·67-s − 2·69-s + 2·71-s + 7·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s − 1.50·11-s − 1.94·13-s + 1.60·19-s + 0.436·21-s − 0.208·23-s − 0.769·27-s + 0.928·29-s + 1.79·31-s − 1.74·33-s − 0.328·37-s − 2.24·39-s + 0.468·41-s + 1.37·43-s + 1.16·47-s − 6/7·49-s − 0.549·53-s + 1.85·57-s + 0.260·59-s − 0.768·61-s + 0.125·63-s + 0.977·67-s − 0.240·69-s + 0.237·71-s + 0.819·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.613999779\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.613999779\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + 7 T + p T^{2} \) | 1.13.h |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 - 17 T + p T^{2} \) | 1.83.ar |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−7.71831071714734749967669958987, −7.49716889440249925296261921311, −6.47639407230893763817108390139, −5.37161327244682819189350995755, −5.02426748471836877599913111170, −4.24044275616014015317231340754, −3.06782596726495254070868412390, −2.70485107836795528854786542143, −2.11927513276796917490181742166, −0.70254069693534336494776706709,
0.70254069693534336494776706709, 2.11927513276796917490181742166, 2.70485107836795528854786542143, 3.06782596726495254070868412390, 4.24044275616014015317231340754, 5.02426748471836877599913111170, 5.37161327244682819189350995755, 6.47639407230893763817108390139, 7.49716889440249925296261921311, 7.71831071714734749967669958987
