| L(s) = 1 | + 2·5-s − 2·7-s − 3·9-s + 6·11-s − 13-s + 17-s + 19-s + 3·23-s − 25-s − 6·29-s + 9·31-s − 4·35-s − 37-s + 3·41-s + 3·43-s − 6·45-s + 6·47-s − 3·49-s + 6·53-s + 12·55-s − 5·59-s + 7·61-s + 6·63-s − 2·65-s − 3·67-s − 8·71-s − 6·73-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.755·7-s − 9-s + 1.80·11-s − 0.277·13-s + 0.242·17-s + 0.229·19-s + 0.625·23-s − 1/5·25-s − 1.11·29-s + 1.61·31-s − 0.676·35-s − 0.164·37-s + 0.468·41-s + 0.457·43-s − 0.894·45-s + 0.875·47-s − 3/7·49-s + 0.824·53-s + 1.61·55-s − 0.650·59-s + 0.896·61-s + 0.755·63-s − 0.248·65-s − 0.366·67-s − 0.949·71-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.119079743\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.119079743\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| 19 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 3 T + p T^{2} \) | 1.43.ad |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 5 T + p T^{2} \) | 1.59.f |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−8.845047563127073625933351501403, −7.64904269614290806124073532039, −6.84482249745042340603670162937, −6.08415882890512186385269517373, −5.82115287805185167947594015246, −4.71449282224709373801940183997, −3.72709382928075820880460285470, −2.97435218403588600481804486295, −2.00882242520336247099209519411, −0.849287523239689284589233499708,
0.849287523239689284589233499708, 2.00882242520336247099209519411, 2.97435218403588600481804486295, 3.72709382928075820880460285470, 4.71449282224709373801940183997, 5.82115287805185167947594015246, 6.08415882890512186385269517373, 6.84482249745042340603670162937, 7.64904269614290806124073532039, 8.845047563127073625933351501403
