| L(s) = 1 | − 8·4-s + 19·13-s + 64·16-s − 56·19-s − 125·25-s + 19·31-s + 323·37-s + 449·43-s − 152·52-s + 901·61-s − 512·64-s − 127·67-s − 1.19e3·73-s + 448·76-s − 1.38e3·79-s − 1.85e3·97-s + 1.00e3·100-s + 1.80e3·103-s − 1.56e3·109-s + ⋯ |
| L(s) = 1 | − 4-s + 0.405·13-s + 16-s − 0.676·19-s − 25-s + 0.110·31-s + 1.43·37-s + 1.59·43-s − 0.405·52-s + 1.89·61-s − 64-s − 0.231·67-s − 1.90·73-s + 0.676·76-s − 1.97·79-s − 1.93·97-s + 100-s + 1.72·103-s − 1.37·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + p^{3} T^{2} \) |
| 5 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 - 19 T + p^{3} T^{2} \) |
| 17 | \( 1 + p^{3} T^{2} \) |
| 19 | \( 1 + 56 T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 + p^{3} T^{2} \) |
| 31 | \( 1 - 19 T + p^{3} T^{2} \) |
| 37 | \( 1 - 323 T + p^{3} T^{2} \) |
| 41 | \( 1 + p^{3} T^{2} \) |
| 43 | \( 1 - 449 T + p^{3} T^{2} \) |
| 47 | \( 1 + p^{3} T^{2} \) |
| 53 | \( 1 + p^{3} T^{2} \) |
| 59 | \( 1 + p^{3} T^{2} \) |
| 61 | \( 1 - 901 T + p^{3} T^{2} \) |
| 67 | \( 1 + 127 T + p^{3} T^{2} \) |
| 71 | \( 1 + p^{3} T^{2} \) |
| 73 | \( 1 + 1190 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1387 T + p^{3} T^{2} \) |
| 83 | \( 1 + p^{3} T^{2} \) |
| 89 | \( 1 + p^{3} T^{2} \) |
| 97 | \( 1 + 1853 T + p^{3} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.850746989610374529340756997674, −8.177443565660214223348556001663, −7.37666593861418284848622447591, −6.18022020726025734049669682416, −5.52870770451373846717983747056, −4.41080149707258431533748763936, −3.86764200468706375188812516001, −2.59641993964101375182365233153, −1.18936224307377599572918504579, 0,
1.18936224307377599572918504579, 2.59641993964101375182365233153, 3.86764200468706375188812516001, 4.41080149707258431533748763936, 5.52870770451373846717983747056, 6.18022020726025734049669682416, 7.37666593861418284848622447591, 8.177443565660214223348556001663, 8.850746989610374529340756997674
