| L(s) = 1 | − 3·9-s + 4·11-s − 13-s + 6·17-s − 4·19-s + 2·29-s − 4·31-s − 6·37-s − 6·41-s + 8·43-s − 7·49-s + 2·53-s − 4·59-s + 10·61-s + 12·67-s − 4·71-s − 14·73-s − 16·79-s + 9·81-s + 12·83-s + 2·89-s + 2·97-s − 12·99-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 9-s + 1.20·11-s − 0.277·13-s + 1.45·17-s − 0.917·19-s + 0.371·29-s − 0.718·31-s − 0.986·37-s − 0.937·41-s + 1.21·43-s − 49-s + 0.274·53-s − 0.520·59-s + 1.28·61-s + 1.46·67-s − 0.474·71-s − 1.63·73-s − 1.80·79-s + 81-s + 1.31·83-s + 0.211·89-s + 0.203·97-s − 1.20·99-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 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.ak |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−16.03076584767465, −15.16064124212409, −14.66136812890578, −14.29668452657629, −13.97973060510750, −13.08810961930627, −12.54852170457297, −11.95596971473204, −11.64064705319712, −10.96729313922718, −10.34391449854718, −9.783771916956431, −9.117766829887316, −8.639965905024436, −8.113404733707942, −7.378218507766687, −6.743588274978109, −6.140493870477550, −5.551297015638893, −4.962313317526125, −4.053312015909508, −3.513585863746641, −2.821574062761505, −1.927160774583427, −1.104601996621796, 0,
1.104601996621796, 1.927160774583427, 2.821574062761505, 3.513585863746641, 4.053312015909508, 4.962313317526125, 5.551297015638893, 6.140493870477550, 6.743588274978109, 7.378218507766687, 8.113404733707942, 8.639965905024436, 9.117766829887316, 9.783771916956431, 10.34391449854718, 10.96729313922718, 11.64064705319712, 11.95596971473204, 12.54852170457297, 13.08810961930627, 13.97973060510750, 14.29668452657629, 14.66136812890578, 15.16064124212409, 16.03076584767465
