| L(s) = 1 | − 3-s − 7-s − 2·9-s − 3·11-s − 4·13-s − 6·17-s + 2·19-s + 21-s − 6·23-s − 5·25-s + 5·27-s − 6·29-s + 4·31-s + 3·33-s + 4·39-s − 9·41-s + 8·43-s + 3·47-s − 6·49-s + 6·51-s + 3·53-s − 2·57-s + 12·59-s + 8·61-s + 2·63-s + 4·67-s + 6·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s − 2/3·9-s − 0.904·11-s − 1.10·13-s − 1.45·17-s + 0.458·19-s + 0.218·21-s − 1.25·23-s − 25-s + 0.962·27-s − 1.11·29-s + 0.718·31-s + 0.522·33-s + 0.640·39-s − 1.40·41-s + 1.21·43-s + 0.437·47-s − 6/7·49-s + 0.840·51-s + 0.412·53-s − 0.264·57-s + 1.56·59-s + 1.02·61-s + 0.251·63-s + 0.488·67-s + 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 37 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−13.90289006288597, −13.77312502410955, −13.07580671099695, −12.69701964877884, −12.06556402889127, −11.65039422219085, −11.30521607660804, −10.68209114483386, −10.15220898918839, −9.754870097711789, −9.258444581030318, −8.506210592469357, −8.154236809090080, −7.502937705525436, −7.012263540024008, −6.412949172446708, −5.904316383154309, −5.310570587178110, −5.030432983957763, −4.182611861304814, −3.746036856800577, −2.756612576046580, −2.435467837826396, −1.793642793026540, −0.5049513384472028, 0,
0.5049513384472028, 1.793642793026540, 2.435467837826396, 2.756612576046580, 3.746036856800577, 4.182611861304814, 5.030432983957763, 5.310570587178110, 5.904316383154309, 6.412949172446708, 7.012263540024008, 7.502937705525436, 8.154236809090080, 8.506210592469357, 9.258444581030318, 9.754870097711789, 10.15220898918839, 10.68209114483386, 11.30521607660804, 11.65039422219085, 12.06556402889127, 12.69701964877884, 13.07580671099695, 13.77312502410955, 13.90289006288597
