| L(s) = 1 | + 3-s − 2·7-s + 9-s − 2·11-s − 13-s + 2·17-s − 6·19-s − 2·21-s − 6·23-s + 27-s − 6·29-s − 2·33-s − 10·37-s − 39-s + 6·41-s − 8·43-s + 6·47-s − 3·49-s + 2·51-s + 10·53-s − 6·57-s − 6·59-s − 10·61-s − 2·63-s − 12·67-s − 6·69-s − 12·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.603·11-s − 0.277·13-s + 0.485·17-s − 1.37·19-s − 0.436·21-s − 1.25·23-s + 0.192·27-s − 1.11·29-s − 0.348·33-s − 1.64·37-s − 0.160·39-s + 0.937·41-s − 1.21·43-s + 0.875·47-s − 3/7·49-s + 0.280·51-s + 1.37·53-s − 0.794·57-s − 0.781·59-s − 1.28·61-s − 0.251·63-s − 1.46·67-s − 0.722·69-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−13.89812802348091, −13.41359552911885, −13.22485896915704, −12.56646526903320, −12.09224540425311, −11.89379366343704, −10.79422895563373, −10.64386049307164, −10.13189765951293, −9.624661400152717, −9.139610256950920, −8.623725986562684, −8.167695753485554, −7.572388592734999, −7.205316520483651, −6.574904566586799, −5.987059836925660, −5.597433851770375, −4.883796017337666, −4.150662719258570, −3.857802670910548, −3.081026206852519, −2.673803369174158, −1.943480779701256, −1.446349703536714, 0, 0,
1.446349703536714, 1.943480779701256, 2.673803369174158, 3.081026206852519, 3.857802670910548, 4.150662719258570, 4.883796017337666, 5.597433851770375, 5.987059836925660, 6.574904566586799, 7.205316520483651, 7.572388592734999, 8.167695753485554, 8.623725986562684, 9.139610256950920, 9.624661400152717, 10.13189765951293, 10.64386049307164, 10.79422895563373, 11.89379366343704, 12.09224540425311, 12.56646526903320, 13.22485896915704, 13.41359552911885, 13.89812802348091
