| L(s) = 1 | − 3-s + 5·7-s + 9-s − 6·11-s + 3·13-s + 2·17-s + 19-s − 5·21-s + 2·23-s − 27-s + 6·29-s + 3·31-s + 6·33-s + 6·37-s − 3·39-s + 4·41-s − 11·43-s + 10·47-s + 18·49-s − 2·51-s + 8·53-s − 57-s − 6·59-s + 3·61-s + 5·63-s + 67-s − 2·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.88·7-s + 1/3·9-s − 1.80·11-s + 0.832·13-s + 0.485·17-s + 0.229·19-s − 1.09·21-s + 0.417·23-s − 0.192·27-s + 1.11·29-s + 0.538·31-s + 1.04·33-s + 0.986·37-s − 0.480·39-s + 0.624·41-s − 1.67·43-s + 1.45·47-s + 18/7·49-s − 0.280·51-s + 1.09·53-s − 0.132·57-s − 0.781·59-s + 0.384·61-s + 0.629·63-s + 0.122·67-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.451917984\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.451917984\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 3 T + p T^{2} \) | 1.61.ad |
| 67 | \( 1 - T + p T^{2} \) | 1.67.ab |
| 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.i |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.71514800896223847283548886009, −10.16661209353267083944596664514, −8.641253522477557635486547756608, −8.022289783499272458145318780682, −7.29517814752026960375125551858, −5.84421768602249075153408308454, −5.14695343031582756584468004415, −4.37818583208833284743712088491, −2.65901886268629700489507936013, −1.20313836023043615938308603780,
1.20313836023043615938308603780, 2.65901886268629700489507936013, 4.37818583208833284743712088491, 5.14695343031582756584468004415, 5.84421768602249075153408308454, 7.29517814752026960375125551858, 8.022289783499272458145318780682, 8.641253522477557635486547756608, 10.16661209353267083944596664514, 10.71514800896223847283548886009
