| L(s) = 1 | + 2-s + 4-s + 5-s + 2·7-s + 8-s + 10-s + 4·11-s − 5·13-s + 2·14-s + 16-s + 17-s + 4·19-s + 20-s + 4·22-s + 6·23-s − 4·25-s − 5·26-s + 2·28-s + 9·29-s − 2·31-s + 32-s + 34-s + 2·35-s − 5·37-s + 4·38-s + 40-s − 2·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.755·7-s + 0.353·8-s + 0.316·10-s + 1.20·11-s − 1.38·13-s + 0.534·14-s + 1/4·16-s + 0.242·17-s + 0.917·19-s + 0.223·20-s + 0.852·22-s + 1.25·23-s − 4/5·25-s − 0.980·26-s + 0.377·28-s + 1.67·29-s − 0.359·31-s + 0.176·32-s + 0.171·34-s + 0.338·35-s − 0.821·37-s + 0.648·38-s + 0.158·40-s − 0.304·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 272322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.353826142\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.353826142\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 41 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 13 T + p T^{2} \) | 1.89.an |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−12.75804602693203, −12.28710431946830, −11.80813747095628, −11.57500394019296, −11.17170703631185, −10.44217831841791, −9.917725262348422, −9.768708849717727, −9.032058813846292, −8.661983773770376, −7.994380734007464, −7.552904675506013, −6.957483138651160, −6.723147107783971, −6.135989611327752, −5.398933339069723, −5.134674556061724, −4.719664780085600, −4.180834912624535, −3.445332734359215, −3.117552517256766, −2.306092105007808, −1.922720437418866, −1.224765672757823, −0.6725068741080174,
0.6725068741080174, 1.224765672757823, 1.922720437418866, 2.306092105007808, 3.117552517256766, 3.445332734359215, 4.180834912624535, 4.719664780085600, 5.134674556061724, 5.398933339069723, 6.135989611327752, 6.723147107783971, 6.957483138651160, 7.552904675506013, 7.994380734007464, 8.661983773770376, 9.032058813846292, 9.768708849717727, 9.917725262348422, 10.44217831841791, 11.17170703631185, 11.57500394019296, 11.80813747095628, 12.28710431946830, 12.75804602693203
