| L(s) = 1 | + 5-s − 5·13-s + 7·17-s − 8·19-s + 23-s − 4·25-s + 10·29-s − 2·31-s + 4·37-s − 8·41-s + 12·43-s + 3·47-s + 9·53-s + 12·59-s + 12·61-s − 5·65-s − 7·67-s − 5·71-s + 73-s − 4·83-s + 7·85-s − 6·89-s − 8·95-s + 10·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.38·13-s + 1.69·17-s − 1.83·19-s + 0.208·23-s − 4/5·25-s + 1.85·29-s − 0.359·31-s + 0.657·37-s − 1.24·41-s + 1.82·43-s + 0.437·47-s + 1.23·53-s + 1.56·59-s + 1.53·61-s − 0.620·65-s − 0.855·67-s − 0.593·71-s + 0.117·73-s − 0.439·83-s + 0.759·85-s − 0.635·89-s − 0.820·95-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.448836225\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.448836225\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−13.14211579467333, −12.74802737758812, −12.39609254372569, −11.79813092300842, −11.60809932143298, −10.67860762819920, −10.27748008854051, −10.03496645989167, −9.606216790394671, −8.835123784540492, −8.549363984678649, −7.896627233440130, −7.458345935284868, −6.950897336943236, −6.379218426535618, −5.869451366529965, −5.352471446006107, −4.906496592816361, −4.182349836537026, −3.851791332662894, −2.914128679853562, −2.487693120005480, −2.020005885684135, −1.141978935372837, −0.4840794335730524,
0.4840794335730524, 1.141978935372837, 2.020005885684135, 2.487693120005480, 2.914128679853562, 3.851791332662894, 4.182349836537026, 4.906496592816361, 5.352471446006107, 5.869451366529965, 6.379218426535618, 6.950897336943236, 7.458345935284868, 7.896627233440130, 8.549363984678649, 8.835123784540492, 9.606216790394671, 10.03496645989167, 10.27748008854051, 10.67860762819920, 11.60809932143298, 11.79813092300842, 12.39609254372569, 12.74802737758812, 13.14211579467333
