| L(s) = 1 | − 3·9-s + 11-s − 2·13-s − 6·17-s + 4·19-s + 4·23-s + 6·29-s + 8·31-s + 2·37-s + 2·41-s + 4·43-s − 12·47-s − 7·49-s + 2·53-s − 4·59-s − 10·61-s − 16·67-s − 8·71-s − 14·73-s − 8·79-s + 9·81-s − 4·83-s + 10·89-s − 10·97-s − 3·99-s − 10·101-s − 4·103-s + ⋯ |
| L(s) = 1 | − 9-s + 0.301·11-s − 0.554·13-s − 1.45·17-s + 0.917·19-s + 0.834·23-s + 1.11·29-s + 1.43·31-s + 0.328·37-s + 0.312·41-s + 0.609·43-s − 1.75·47-s − 49-s + 0.274·53-s − 0.520·59-s − 1.28·61-s − 1.95·67-s − 0.949·71-s − 1.63·73-s − 0.900·79-s + 81-s − 0.439·83-s + 1.05·89-s − 1.01·97-s − 0.301·99-s − 0.995·101-s − 0.394·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 16 T + p T^{2} \) | 1.67.q |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−8.067324706031938881269001188358, −7.25409359989160561546791965971, −6.48067063437882788912913410261, −5.91522945024739734859341901763, −4.84786612010371112418366573978, −4.45768242969291673579097656708, −3.07429403089419103076246288907, −2.69402268918089176010114790698, −1.35944850033720564570853590086, 0,
1.35944850033720564570853590086, 2.69402268918089176010114790698, 3.07429403089419103076246288907, 4.45768242969291673579097656708, 4.84786612010371112418366573978, 5.91522945024739734859341901763, 6.48067063437882788912913410261, 7.25409359989160561546791965971, 8.067324706031938881269001188358
