| L(s) = 1 | + 2·3-s − 5-s − 4·7-s + 9-s + 11-s − 2·15-s + 4·17-s − 4·19-s − 8·21-s − 6·23-s + 25-s − 4·27-s − 10·29-s + 4·31-s + 2·33-s + 4·35-s − 2·37-s + 10·41-s + 8·43-s − 45-s − 6·47-s + 9·49-s + 8·51-s + 2·53-s − 55-s − 8·57-s − 4·59-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s + 0.301·11-s − 0.516·15-s + 0.970·17-s − 0.917·19-s − 1.74·21-s − 1.25·23-s + 1/5·25-s − 0.769·27-s − 1.85·29-s + 0.718·31-s + 0.348·33-s + 0.676·35-s − 0.328·37-s + 1.56·41-s + 1.21·43-s − 0.149·45-s − 0.875·47-s + 9/7·49-s + 1.12·51-s + 0.274·53-s − 0.134·55-s − 1.05·57-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 297440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 12 T + p T^{2} \) | 1.73.m |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.83742043464394, −12.62784321024834, −12.15901868161798, −11.59692463001432, −11.11583626162721, −10.47193945694601, −10.06364306971724, −9.611044911635007, −9.172610191805491, −8.931140783420995, −8.267951332006894, −7.838847454647303, −7.422546693844567, −7.041501021989752, −6.196531354915884, −5.985383407733674, −5.588912227459548, −4.474007009054222, −4.193708041713317, −3.602346487752931, −3.250188508139208, −2.820225537859397, −2.158361916655665, −1.650680337206432, −0.6409544021284036, 0,
0.6409544021284036, 1.650680337206432, 2.158361916655665, 2.820225537859397, 3.250188508139208, 3.602346487752931, 4.193708041713317, 4.474007009054222, 5.588912227459548, 5.985383407733674, 6.196531354915884, 7.041501021989752, 7.422546693844567, 7.838847454647303, 8.267951332006894, 8.931140783420995, 9.172610191805491, 9.611044911635007, 10.06364306971724, 10.47193945694601, 11.11583626162721, 11.59692463001432, 12.15901868161798, 12.62784321024834, 12.83742043464394
