| L(s) = 1 | − 3-s + 5-s + 4·7-s + 9-s − 6·11-s + 2·13-s − 15-s − 6·17-s − 2·19-s − 4·21-s + 25-s − 27-s + 6·29-s − 8·31-s + 6·33-s + 4·35-s + 37-s − 2·39-s − 6·41-s − 8·43-s + 45-s − 6·47-s + 9·49-s + 6·51-s + 6·53-s − 6·55-s + 2·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s − 1.80·11-s + 0.554·13-s − 0.258·15-s − 1.45·17-s − 0.458·19-s − 0.872·21-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 1.04·33-s + 0.676·35-s + 0.164·37-s − 0.320·39-s − 0.937·41-s − 1.21·43-s + 0.149·45-s − 0.875·47-s + 9/7·49-s + 0.840·51-s + 0.824·53-s − 0.809·55-s + 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.677633988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.677633988\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 37 | \( 1 - T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 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 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−7.88490674930624237986040718133, −6.96165291772688103919348452577, −6.42281917887630762159396976932, −5.40209409240129424016558382093, −5.10773611823564971575397049931, −4.54819374963735396242901787561, −3.53320746341258620082863360917, −2.23988396323307688436465491180, −1.95351103406954527954573674898, −0.63181959746510107924075287826,
0.63181959746510107924075287826, 1.95351103406954527954573674898, 2.23988396323307688436465491180, 3.53320746341258620082863360917, 4.54819374963735396242901787561, 5.10773611823564971575397049931, 5.40209409240129424016558382093, 6.42281917887630762159396976932, 6.96165291772688103919348452577, 7.88490674930624237986040718133
