| L(s) = 1 | + 2·5-s + 2·7-s + 6·13-s + 6·23-s − 25-s − 6·29-s − 10·31-s + 4·35-s + 2·37-s + 4·43-s − 8·47-s − 3·49-s + 6·53-s + 10·61-s + 12·65-s − 8·67-s + 10·71-s + 16·73-s + 6·79-s + 16·83-s − 10·89-s + 12·91-s − 12·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s + 1.66·13-s + 1.25·23-s − 1/5·25-s − 1.11·29-s − 1.79·31-s + 0.676·35-s + 0.328·37-s + 0.609·43-s − 1.16·47-s − 3/7·49-s + 0.824·53-s + 1.28·61-s + 1.48·65-s − 0.977·67-s + 1.18·71-s + 1.87·73-s + 0.675·79-s + 1.75·83-s − 1.05·89-s + 1.25·91-s − 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 166464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.247911015\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.247911015\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 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.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−13.33311167777160, −12.90334045621539, −12.42007966450126, −11.67364243843078, −11.15312199595946, −10.91811932157977, −10.65390278749966, −9.700991078984493, −9.441370779520469, −9.021411473639578, −8.390609872466330, −8.063043964182997, −7.425429598145162, −6.834531778301024, −6.383039209024185, −5.798868903833779, −5.286730476878928, −5.107954714680828, −4.069126622545982, −3.788191623063833, −3.144000046202223, −2.336622060783518, −1.764680628104603, −1.370841220770718, −0.6034468103462076,
0.6034468103462076, 1.370841220770718, 1.764680628104603, 2.336622060783518, 3.144000046202223, 3.788191623063833, 4.069126622545982, 5.107954714680828, 5.286730476878928, 5.798868903833779, 6.383039209024185, 6.834531778301024, 7.425429598145162, 8.063043964182997, 8.390609872466330, 9.021411473639578, 9.441370779520469, 9.700991078984493, 10.65390278749966, 10.91811932157977, 11.15312199595946, 11.67364243843078, 12.42007966450126, 12.90334045621539, 13.33311167777160
