| L(s) = 1 | − 3-s + 9-s − 4·11-s − 2·17-s + 6·19-s + 6·23-s − 27-s + 2·29-s + 2·31-s + 4·33-s + 6·37-s − 2·41-s − 4·43-s + 8·47-s − 7·49-s + 2·51-s − 10·53-s − 6·57-s + 4·59-s − 10·61-s − 12·67-s − 6·69-s + 8·71-s + 12·73-s + 2·79-s + 81-s + 10·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.20·11-s − 0.485·17-s + 1.37·19-s + 1.25·23-s − 0.192·27-s + 0.371·29-s + 0.359·31-s + 0.696·33-s + 0.986·37-s − 0.312·41-s − 0.609·43-s + 1.16·47-s − 49-s + 0.280·51-s − 1.37·53-s − 0.794·57-s + 0.520·59-s − 1.28·61-s − 1.46·67-s − 0.722·69-s + 0.949·71-s + 1.40·73-s + 0.225·79-s + 1/9·81-s + 1.09·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 339600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 339600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 283 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 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
−12.73932099121433, −12.35850244480506, −11.89142650530630, −11.35581370973009, −11.02419286937081, −10.61362201194316, −10.13552449629548, −9.675376412146350, −9.174098119493754, −8.808453304732404, −7.979195055752079, −7.752462689302225, −7.373064983293150, −6.539709677748598, −6.475492882726283, −5.686717027518459, −5.241624739745238, −4.818403477879497, −4.547254569787029, −3.625386850017885, −3.163247806570368, −2.673566554845818, −2.065182670373295, −1.256750247456205, −0.7540382641133918, 0,
0.7540382641133918, 1.256750247456205, 2.065182670373295, 2.673566554845818, 3.163247806570368, 3.625386850017885, 4.547254569787029, 4.818403477879497, 5.241624739745238, 5.686717027518459, 6.475492882726283, 6.539709677748598, 7.373064983293150, 7.752462689302225, 7.979195055752079, 8.808453304732404, 9.174098119493754, 9.675376412146350, 10.13552449629548, 10.61362201194316, 11.02419286937081, 11.35581370973009, 11.89142650530630, 12.35850244480506, 12.73932099121433
