| L(s) = 1 | + 3-s + 9-s − 2·13-s − 2·17-s − 8·19-s − 23-s + 27-s + 2·29-s + 8·31-s + 2·37-s − 2·39-s + 10·41-s − 8·43-s + 8·47-s − 7·49-s − 2·51-s + 2·53-s − 8·57-s − 4·59-s − 2·61-s − 8·67-s − 69-s + 6·73-s − 8·79-s + 81-s + 16·83-s + 2·87-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.554·13-s − 0.485·17-s − 1.83·19-s − 0.208·23-s + 0.192·27-s + 0.371·29-s + 1.43·31-s + 0.328·37-s − 0.320·39-s + 1.56·41-s − 1.21·43-s + 1.16·47-s − 49-s − 0.280·51-s + 0.274·53-s − 1.05·57-s − 0.520·59-s − 0.256·61-s − 0.977·67-s − 0.120·69-s + 0.702·73-s − 0.900·79-s + 1/9·81-s + 1.75·83-s + 0.214·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−13.89348985831431, −13.38273143458125, −13.01728742898509, −12.42468268341172, −12.11836842389006, −11.43641621026673, −10.94010264307937, −10.32016418943020, −10.11628932336835, −9.392358766110350, −8.880867601817347, −8.574927759291553, −7.844621986618544, −7.647460373373899, −6.831688506141161, −6.294873931589044, −6.099371739524253, −5.028821397816659, −4.646823511755937, −4.156536872877593, −3.567327647091574, −2.716127811765410, −2.402379623614299, −1.766816078807569, −0.8716523651615348, 0,
0.8716523651615348, 1.766816078807569, 2.402379623614299, 2.716127811765410, 3.567327647091574, 4.156536872877593, 4.646823511755937, 5.028821397816659, 6.099371739524253, 6.294873931589044, 6.831688506141161, 7.647460373373899, 7.844621986618544, 8.574927759291553, 8.880867601817347, 9.392358766110350, 10.11628932336835, 10.32016418943020, 10.94010264307937, 11.43641621026673, 12.11836842389006, 12.42468268341172, 13.01728742898509, 13.38273143458125, 13.89348985831431
