| L(s) = 1 | − 2·3-s − 7-s + 9-s − 13-s − 19-s + 2·21-s − 9·23-s + 4·27-s + 3·29-s − 5·31-s + 4·37-s + 2·39-s + 43-s + 49-s + 6·53-s + 2·57-s − 2·61-s − 63-s + 2·67-s + 18·69-s + 9·71-s − 4·73-s − 4·79-s − 11·81-s + 9·83-s − 6·87-s + 15·89-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.377·7-s + 1/3·9-s − 0.277·13-s − 0.229·19-s + 0.436·21-s − 1.87·23-s + 0.769·27-s + 0.557·29-s − 0.898·31-s + 0.657·37-s + 0.320·39-s + 0.152·43-s + 1/7·49-s + 0.824·53-s + 0.264·57-s − 0.256·61-s − 0.125·63-s + 0.244·67-s + 2.16·69-s + 1.06·71-s − 0.468·73-s − 0.450·79-s − 1.22·81-s + 0.987·83-s − 0.643·87-s + 1.58·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−12.64232308389176, −12.17709015576505, −11.97944181417337, −11.53540645086241, −10.93608511839265, −10.61226351234484, −10.20627166833187, −9.666011895643594, −9.326327151902283, −8.650136403149516, −8.169434237194125, −7.751650355570402, −7.079424503649335, −6.730486300569527, −6.147132297413177, −5.812944758411519, −5.489601906803911, −4.707851976178425, −4.477158318165978, −3.728164346382606, −3.339043765873127, −2.427834852690062, −2.139323643478708, −1.254435200031817, −0.5729312749207969, 0,
0.5729312749207969, 1.254435200031817, 2.139323643478708, 2.427834852690062, 3.339043765873127, 3.728164346382606, 4.477158318165978, 4.707851976178425, 5.489601906803911, 5.812944758411519, 6.147132297413177, 6.730486300569527, 7.079424503649335, 7.751650355570402, 8.169434237194125, 8.650136403149516, 9.326327151902283, 9.666011895643594, 10.20627166833187, 10.61226351234484, 10.93608511839265, 11.53540645086241, 11.97944181417337, 12.17709015576505, 12.64232308389176
