| L(s) = 1 | + 4·7-s + 4·13-s + 8·19-s + 4·23-s + 6·29-s + 8·31-s − 4·37-s − 6·41-s − 4·43-s − 4·47-s + 9·49-s + 12·53-s − 6·61-s − 12·67-s − 16·71-s + 8·79-s − 12·83-s + 10·89-s + 16·91-s + 8·97-s − 14·101-s + 12·103-s + 12·107-s − 10·109-s + 8·113-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 1.10·13-s + 1.83·19-s + 0.834·23-s + 1.11·29-s + 1.43·31-s − 0.657·37-s − 0.937·41-s − 0.609·43-s − 0.583·47-s + 9/7·49-s + 1.64·53-s − 0.768·61-s − 1.46·67-s − 1.89·71-s + 0.900·79-s − 1.31·83-s + 1.05·89-s + 1.67·91-s + 0.812·97-s − 1.39·101-s + 1.18·103-s + 1.16·107-s − 0.957·109-s + 0.752·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.114270605\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.114270605\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−7.944512322026164068114502692709, −7.34269050459558939946490767184, −6.56622895641247859679189760888, −5.71777206241947850918125172642, −5.00664050322097196617773387555, −4.55255081555298125996070677298, −3.50056447948240150642063671920, −2.78534335651101145386977301540, −1.53337136723193171388562354438, −1.03818003112559218833844009438,
1.03818003112559218833844009438, 1.53337136723193171388562354438, 2.78534335651101145386977301540, 3.50056447948240150642063671920, 4.55255081555298125996070677298, 5.00664050322097196617773387555, 5.71777206241947850918125172642, 6.56622895641247859679189760888, 7.34269050459558939946490767184, 7.944512322026164068114502692709
