| L(s) = 1 | + 3-s + 5-s + 9-s − 2·13-s + 15-s + 6·17-s + 4·23-s + 25-s + 27-s + 2·29-s − 8·31-s − 6·37-s − 2·39-s + 6·41-s + 12·43-s + 45-s − 12·47-s + 6·51-s + 10·53-s − 8·59-s − 10·61-s − 2·65-s − 12·67-s + 4·69-s − 8·71-s − 10·73-s + 75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.554·13-s + 0.258·15-s + 1.45·17-s + 0.834·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s − 1.43·31-s − 0.986·37-s − 0.320·39-s + 0.937·41-s + 1.82·43-s + 0.149·45-s − 1.75·47-s + 0.840·51-s + 1.37·53-s − 1.04·59-s − 1.28·61-s − 0.248·65-s − 1.46·67-s + 0.481·69-s − 0.949·71-s − 1.17·73-s + 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 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.i |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−14.70947252567169, −14.37859858394668, −13.97572791403876, −13.25526978886749, −12.88688448714120, −12.30698103925290, −11.94706925385350, −11.12165360565988, −10.63342284281648, −10.07110493075069, −9.674516523325726, −8.914140688664579, −8.838134125472228, −7.861673147674799, −7.387265097159665, −7.134213869636631, −6.156329594088578, −5.700584988800480, −5.121557601198655, −4.450804278278625, −3.785507452647063, −2.961859457999929, −2.740597529085307, −1.661456297230699, −1.233595550104768, 0,
1.233595550104768, 1.661456297230699, 2.740597529085307, 2.961859457999929, 3.785507452647063, 4.450804278278625, 5.121557601198655, 5.700584988800480, 6.156329594088578, 7.134213869636631, 7.387265097159665, 7.861673147674799, 8.838134125472228, 8.914140688664579, 9.674516523325726, 10.07110493075069, 10.63342284281648, 11.12165360565988, 11.94706925385350, 12.30698103925290, 12.88688448714120, 13.25526978886749, 13.97572791403876, 14.37859858394668, 14.70947252567169
