| L(s) = 1 | − 3-s − 2·4-s + 5-s + 2·7-s + 9-s − 11-s + 2·12-s + 6·13-s − 15-s + 4·16-s − 2·17-s − 3·19-s − 2·20-s − 2·21-s + 25-s − 27-s − 4·28-s − 10·29-s + 3·31-s + 33-s + 2·35-s − 2·36-s − 4·37-s − 6·39-s + 9·41-s − 8·43-s + 2·44-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.301·11-s + 0.577·12-s + 1.66·13-s − 0.258·15-s + 16-s − 0.485·17-s − 0.688·19-s − 0.447·20-s − 0.436·21-s + 1/5·25-s − 0.192·27-s − 0.755·28-s − 1.85·29-s + 0.538·31-s + 0.174·33-s + 0.338·35-s − 1/3·36-s − 0.657·37-s − 0.960·39-s + 1.40·41-s − 1.21·43-s + 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 23 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−7.70892941811742750035482045719, −6.56859197464400092845592361059, −6.03655875574003110145676089280, −5.36452344203324969546062015039, −4.74538483287744760732265255104, −4.05583597520259383443101581479, −3.33422466314695884283330111097, −1.94575786510579960038879915931, −1.22715850795367572215790688243, 0,
1.22715850795367572215790688243, 1.94575786510579960038879915931, 3.33422466314695884283330111097, 4.05583597520259383443101581479, 4.74538483287744760732265255104, 5.36452344203324969546062015039, 6.03655875574003110145676089280, 6.56859197464400092845592361059, 7.70892941811742750035482045719
