| L(s) = 1 | − 7-s + 11-s − 2·13-s + 6·19-s − 23-s − 29-s − 2·31-s − 7·37-s + 8·41-s − 43-s + 2·47-s + 49-s + 14·53-s − 10·59-s − 3·67-s + 9·71-s − 77-s + 79-s − 2·83-s − 2·89-s + 2·91-s − 10·97-s + 12·101-s + 2·103-s + 4·107-s + 9·109-s + 9·113-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.301·11-s − 0.554·13-s + 1.37·19-s − 0.208·23-s − 0.185·29-s − 0.359·31-s − 1.15·37-s + 1.24·41-s − 0.152·43-s + 0.291·47-s + 1/7·49-s + 1.92·53-s − 1.30·59-s − 0.366·67-s + 1.06·71-s − 0.113·77-s + 0.112·79-s − 0.219·83-s − 0.211·89-s + 0.209·91-s − 1.01·97-s + 1.19·101-s + 0.197·103-s + 0.386·107-s + 0.862·109-s + 0.846·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.759576273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.759576273\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| good | 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 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
−7.904453357336250542277945978212, −7.31976198943964832414139567805, −6.74659882658412398535553167643, −5.81464669926226680763336847535, −5.29555087040575953756515648194, −4.38360839169442490484249204662, −3.57740299323049360586678381152, −2.83562009643549761862315939653, −1.84629882751458527881304627929, −0.68988872283584156221148626470,
0.68988872283584156221148626470, 1.84629882751458527881304627929, 2.83562009643549761862315939653, 3.57740299323049360586678381152, 4.38360839169442490484249204662, 5.29555087040575953756515648194, 5.81464669926226680763336847535, 6.74659882658412398535553167643, 7.31976198943964832414139567805, 7.904453357336250542277945978212
