| L(s) = 1 | − 3-s + 9-s + 13-s + 6·17-s + 5·19-s − 6·23-s − 27-s + 6·29-s − 5·31-s − 7·37-s − 39-s − 12·41-s − 43-s + 6·47-s − 6·51-s − 5·57-s − 6·59-s + 2·61-s − 7·67-s + 6·69-s + 12·71-s + 11·73-s − 13·79-s + 81-s + 12·83-s − 6·87-s − 6·89-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.277·13-s + 1.45·17-s + 1.14·19-s − 1.25·23-s − 0.192·27-s + 1.11·29-s − 0.898·31-s − 1.15·37-s − 0.160·39-s − 1.87·41-s − 0.152·43-s + 0.875·47-s − 0.840·51-s − 0.662·57-s − 0.781·59-s + 0.256·61-s − 0.855·67-s + 0.722·69-s + 1.42·71-s + 1.28·73-s − 1.46·79-s + 1/9·81-s + 1.31·83-s − 0.643·87-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.642787858\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.642787858\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 + 13 T + p T^{2} \) | 1.79.n |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−12.74484107706657, −12.26812704936073, −12.01535894891594, −11.69267670973248, −11.06623725084188, −10.47996159487636, −10.19967454448256, −9.767454356392552, −9.275737824952027, −8.644677984404324, −8.129748238994721, −7.726639006725928, −7.203185269670969, −6.694364020107718, −6.204074869147486, −5.565651887139086, −5.318924879951796, −4.828495309772213, −4.060923289798511, −3.525908212869003, −3.202676865075210, −2.368985872233843, −1.603427842642705, −1.193264035483055, −0.3903505719195739,
0.3903505719195739, 1.193264035483055, 1.603427842642705, 2.368985872233843, 3.202676865075210, 3.525908212869003, 4.060923289798511, 4.828495309772213, 5.318924879951796, 5.565651887139086, 6.204074869147486, 6.694364020107718, 7.203185269670969, 7.726639006725928, 8.129748238994721, 8.644677984404324, 9.275737824952027, 9.767454356392552, 10.19967454448256, 10.47996159487636, 11.06623725084188, 11.69267670973248, 12.01535894891594, 12.26812704936073, 12.74484107706657
