| L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s + 2·11-s + 6·13-s + 15-s − 6·17-s − 6·19-s + 2·21-s + 2·23-s + 25-s − 27-s + 2·29-s − 4·31-s − 2·33-s + 2·35-s + 10·37-s − 6·39-s − 2·41-s + 8·43-s − 45-s − 6·47-s − 3·49-s + 6·51-s + 6·53-s − 2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.603·11-s + 1.66·13-s + 0.258·15-s − 1.45·17-s − 1.37·19-s + 0.436·21-s + 0.417·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s − 0.718·31-s − 0.348·33-s + 0.338·35-s + 1.64·37-s − 0.960·39-s − 0.312·41-s + 1.21·43-s − 0.149·45-s − 0.875·47-s − 3/7·49-s + 0.840·51-s + 0.824·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 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
−8.894292507833391181972425046495, −8.126071021574085392020093246251, −6.98075497484175451848100649927, −6.36726321994570280187001552458, −5.91219562296196658340104080812, −4.44962269551754541751387325822, −4.03643784924633530793627705394, −2.89366190370110233973521884634, −1.44640173159626840226202689232, 0,
1.44640173159626840226202689232, 2.89366190370110233973521884634, 4.03643784924633530793627705394, 4.44962269551754541751387325822, 5.91219562296196658340104080812, 6.36726321994570280187001552458, 6.98075497484175451848100649927, 8.126071021574085392020093246251, 8.894292507833391181972425046495
