| L(s) = 1 | + 3-s − 7-s + 9-s + 2·11-s − 6·13-s − 17-s − 21-s − 4·23-s + 27-s + 6·29-s + 6·31-s + 2·33-s − 6·37-s − 6·39-s − 6·41-s + 10·43-s − 6·47-s + 49-s − 51-s − 6·53-s + 8·59-s − 2·61-s − 63-s + 10·67-s − 4·69-s + 6·71-s + 10·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.603·11-s − 1.66·13-s − 0.242·17-s − 0.218·21-s − 0.834·23-s + 0.192·27-s + 1.11·29-s + 1.07·31-s + 0.348·33-s − 0.986·37-s − 0.960·39-s − 0.937·41-s + 1.52·43-s − 0.875·47-s + 1/7·49-s − 0.140·51-s − 0.824·53-s + 1.04·59-s − 0.256·61-s − 0.125·63-s + 1.22·67-s − 0.481·69-s + 0.712·71-s + 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 + T \) | |
| good | 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−15.19165595049385, −14.53022415648145, −14.06789893845715, −13.92557826882823, −12.98707644310566, −12.61653184574141, −12.00642097021660, −11.74783081713344, −10.90090925057420, −10.12487916478044, −9.916618777627749, −9.406154653243465, −8.714376345145771, −8.233743032656144, −7.645935410981020, −6.971450005185050, −6.610744246675100, −5.931546785085931, −5.019172484417075, −4.639753915314955, −3.895229552770808, −3.253279257495421, −2.513708604926210, −2.052700355997736, −1.020385482529998, 0,
1.020385482529998, 2.052700355997736, 2.513708604926210, 3.253279257495421, 3.895229552770808, 4.639753915314955, 5.019172484417075, 5.931546785085931, 6.610744246675100, 6.971450005185050, 7.645935410981020, 8.233743032656144, 8.714376345145771, 9.406154653243465, 9.916618777627749, 10.12487916478044, 10.90090925057420, 11.74783081713344, 12.00642097021660, 12.61653184574141, 12.98707644310566, 13.92557826882823, 14.06789893845715, 14.53022415648145, 15.19165595049385
