| L(s) = 1 | + 3-s + 2·5-s + 7-s + 9-s + 11-s − 3·13-s + 2·15-s − 3·17-s + 2·19-s + 21-s + 8·23-s − 25-s + 27-s + 29-s − 8·31-s + 33-s + 2·35-s − 3·39-s + 2·41-s + 2·45-s + 5·47-s − 6·49-s − 3·51-s − 2·53-s + 2·55-s + 2·57-s − 6·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.832·13-s + 0.516·15-s − 0.727·17-s + 0.458·19-s + 0.218·21-s + 1.66·23-s − 1/5·25-s + 0.192·27-s + 0.185·29-s − 1.43·31-s + 0.174·33-s + 0.338·35-s − 0.480·39-s + 0.312·41-s + 0.298·45-s + 0.729·47-s − 6/7·49-s − 0.420·51-s − 0.274·53-s + 0.269·55-s + 0.264·57-s − 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.856994818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.856994818\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 29 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 5 T + p T^{2} \) | 1.47.af |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−11.40895927564426908442159942943, −10.50064536446632234144619752022, −9.426303336015217858684772555140, −8.984797713959564715274282326442, −7.68279371077555725335656773359, −6.81678658339170657785131854385, −5.55463478091332443970021031210, −4.51698377791384209106277246274, −2.97182530915931491923506365555, −1.74153678405777120495362082920,
1.74153678405777120495362082920, 2.97182530915931491923506365555, 4.51698377791384209106277246274, 5.55463478091332443970021031210, 6.81678658339170657785131854385, 7.68279371077555725335656773359, 8.984797713959564715274282326442, 9.426303336015217858684772555140, 10.50064536446632234144619752022, 11.40895927564426908442159942943
