| L(s) = 1 | + 3-s − 7-s + 9-s + 4·11-s + 3·13-s − 8·17-s − 4·19-s − 21-s − 5·25-s + 27-s + 4·29-s + 8·31-s + 4·33-s + 37-s + 3·39-s + 8·41-s + 43-s + 6·47-s − 6·49-s − 8·51-s − 6·53-s − 4·57-s + 6·59-s − 15·61-s − 63-s − 13·67-s + 10·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.832·13-s − 1.94·17-s − 0.917·19-s − 0.218·21-s − 25-s + 0.192·27-s + 0.742·29-s + 1.43·31-s + 0.696·33-s + 0.164·37-s + 0.480·39-s + 1.24·41-s + 0.152·43-s + 0.875·47-s − 6/7·49-s − 1.12·51-s − 0.824·53-s − 0.529·57-s + 0.781·59-s − 1.92·61-s − 0.125·63-s − 1.58·67-s + 1.18·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 101568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.719470464\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.719470464\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 15 T + p T^{2} \) | 1.61.p |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 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
−13.78613073279618, −13.28456960562025, −12.93932217570791, −12.24396263524874, −11.84978305650921, −11.24266153223092, −10.80931804760111, −10.34539057700395, −9.601552856344438, −9.225297674271498, −8.836512009104736, −8.340766592193756, −7.883025201764762, −7.092138589436511, −6.601075546735291, −6.212785626796337, −5.886318996110381, −4.679018442882983, −4.307175192200436, −4.034686650963862, −3.202229365356776, −2.650571307813159, −1.973978895363846, −1.382481141203957, −0.4967684529146761,
0.4967684529146761, 1.382481141203957, 1.973978895363846, 2.650571307813159, 3.202229365356776, 4.034686650963862, 4.307175192200436, 4.679018442882983, 5.886318996110381, 6.212785626796337, 6.601075546735291, 7.092138589436511, 7.883025201764762, 8.340766592193756, 8.836512009104736, 9.225297674271498, 9.601552856344438, 10.34539057700395, 10.80931804760111, 11.24266153223092, 11.84978305650921, 12.24396263524874, 12.93932217570791, 13.28456960562025, 13.78613073279618
