| L(s) = 1 | + 2·5-s + 4·11-s − 13-s − 6·17-s − 4·19-s − 25-s − 4·29-s − 4·31-s − 12·37-s − 12·41-s − 8·43-s + 2·47-s + 8·53-s + 8·55-s + 4·59-s − 10·61-s − 2·65-s − 14·67-s + 8·71-s − 2·73-s − 16·79-s − 12·85-s − 4·89-s − 8·95-s − 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.20·11-s − 0.277·13-s − 1.45·17-s − 0.917·19-s − 1/5·25-s − 0.742·29-s − 0.718·31-s − 1.97·37-s − 1.87·41-s − 1.21·43-s + 0.291·47-s + 1.09·53-s + 1.07·55-s + 0.520·59-s − 1.28·61-s − 0.248·65-s − 1.71·67-s + 0.949·71-s − 0.234·73-s − 1.80·79-s − 1.30·85-s − 0.423·89-s − 0.820·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 366912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 366912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2151069563\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2151069563\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 12 T + p T^{2} \) | 1.37.m |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−12.46638843950244, −11.99549485669724, −11.68799633662101, −11.15185639303442, −10.61516982388781, −10.25555246446744, −9.834152936152989, −9.227656124045616, −8.812141211711406, −8.717336361007632, −8.024098385833364, −7.217261255135470, −6.888842472404412, −6.567952846692014, −6.057229518104605, −5.508642255199888, −5.075066464647139, −4.462322312736330, −3.962607709356408, −3.531020477310652, −2.825574173453986, −2.102743829587664, −1.745996870533828, −1.407766075968603, −0.1078279520424876,
0.1078279520424876, 1.407766075968603, 1.745996870533828, 2.102743829587664, 2.825574173453986, 3.531020477310652, 3.962607709356408, 4.462322312736330, 5.075066464647139, 5.508642255199888, 6.057229518104605, 6.567952846692014, 6.888842472404412, 7.217261255135470, 8.024098385833364, 8.717336361007632, 8.812141211711406, 9.227656124045616, 9.834152936152989, 10.25555246446744, 10.61516982388781, 11.15185639303442, 11.68799633662101, 11.99549485669724, 12.46638843950244
