| L(s) = 1 | − 5-s − 4·11-s − 2·13-s − 2·17-s − 19-s + 25-s − 6·29-s + 6·37-s + 6·41-s + 4·43-s − 8·47-s − 7·49-s − 6·53-s + 4·55-s + 12·59-s − 2·61-s + 2·65-s + 4·67-s − 8·71-s − 6·73-s + 16·79-s + 4·83-s + 2·85-s + 6·89-s + 95-s + 10·97-s + 10·101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.20·11-s − 0.554·13-s − 0.485·17-s − 0.229·19-s + 1/5·25-s − 1.11·29-s + 0.986·37-s + 0.937·41-s + 0.609·43-s − 1.16·47-s − 49-s − 0.824·53-s + 0.539·55-s + 1.56·59-s − 0.256·61-s + 0.248·65-s + 0.488·67-s − 0.949·71-s − 0.702·73-s + 1.80·79-s + 0.439·83-s + 0.216·85-s + 0.635·89-s + 0.102·95-s + 1.01·97-s + 0.995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.053340527\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.053340527\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 19 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−7.80677633361074454034617187587, −7.49680457954522492076762538065, −6.59259259597042389447574016415, −5.84289709807230750803159708395, −5.03390034296621196331900440623, −4.47575681554886547584825793205, −3.55933901609947550174739346667, −2.70610336724282529715906832711, −1.95854307692920622402439334992, −0.50015434389376471625756468565,
0.50015434389376471625756468565, 1.95854307692920622402439334992, 2.70610336724282529715906832711, 3.55933901609947550174739346667, 4.47575681554886547584825793205, 5.03390034296621196331900440623, 5.84289709807230750803159708395, 6.59259259597042389447574016415, 7.49680457954522492076762538065, 7.80677633361074454034617187587
