| L(s) = 1 | + 2-s + 3-s + 4-s + 4·5-s + 6-s + 8-s + 9-s + 4·10-s − 11-s + 12-s + 6·13-s + 4·15-s + 16-s + 4·17-s + 18-s + 2·19-s + 4·20-s − 22-s − 8·23-s + 24-s + 11·25-s + 6·26-s + 27-s − 6·29-s + 4·30-s − 6·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.78·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 1.26·10-s − 0.301·11-s + 0.288·12-s + 1.66·13-s + 1.03·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s + 0.458·19-s + 0.894·20-s − 0.213·22-s − 1.66·23-s + 0.204·24-s + 11/5·25-s + 1.17·26-s + 0.192·27-s − 1.11·29-s + 0.730·30-s − 1.07·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.439384372\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.439384372\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−8.735998128715284492998822871091, −7.907770256544801027606829969998, −7.01388368279354966786295768024, −6.04350352134711634862151937108, −5.78641126273592614924413513429, −4.99001449691342549407846970736, −3.72515787941803042519247102042, −3.18439835936289363592396358072, −1.95397093408402137319315879397, −1.53287618374872470103273275153,
1.53287618374872470103273275153, 1.95397093408402137319315879397, 3.18439835936289363592396358072, 3.72515787941803042519247102042, 4.99001449691342549407846970736, 5.78641126273592614924413513429, 6.04350352134711634862151937108, 7.01388368279354966786295768024, 7.907770256544801027606829969998, 8.735998128715284492998822871091
