L(s) = 1 | − 3·3-s − 18·5-s − 8·7-s + 9·9-s − 36·11-s − 10·13-s + 54·15-s + 18·17-s + 100·19-s + 24·21-s − 72·23-s + 199·25-s − 27·27-s − 234·29-s + 16·31-s + 108·33-s + 144·35-s − 226·37-s + 30·39-s + 90·41-s − 452·43-s − 162·45-s − 432·47-s − 279·49-s − 54·51-s + 414·53-s + 648·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.60·5-s − 0.431·7-s + 1/3·9-s − 0.986·11-s − 0.213·13-s + 0.929·15-s + 0.256·17-s + 1.20·19-s + 0.249·21-s − 0.652·23-s + 1.59·25-s − 0.192·27-s − 1.49·29-s + 0.0926·31-s + 0.569·33-s + 0.695·35-s − 1.00·37-s + 0.123·39-s + 0.342·41-s − 1.60·43-s − 0.536·45-s − 1.34·47-s − 0.813·49-s − 0.148·51-s + 1.07·53-s + 1.58·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
good | 5 | \( 1 + 18 T + p^{3} T^{2} \) |
| 7 | \( 1 + 8 T + p^{3} T^{2} \) |
| 11 | \( 1 + 36 T + p^{3} T^{2} \) |
| 13 | \( 1 + 10 T + p^{3} T^{2} \) |
| 17 | \( 1 - 18 T + p^{3} T^{2} \) |
| 19 | \( 1 - 100 T + p^{3} T^{2} \) |
| 23 | \( 1 + 72 T + p^{3} T^{2} \) |
| 29 | \( 1 + 234 T + p^{3} T^{2} \) |
| 31 | \( 1 - 16 T + p^{3} T^{2} \) |
| 37 | \( 1 + 226 T + p^{3} T^{2} \) |
| 41 | \( 1 - 90 T + p^{3} T^{2} \) |
| 43 | \( 1 + 452 T + p^{3} T^{2} \) |
| 47 | \( 1 + 432 T + p^{3} T^{2} \) |
| 53 | \( 1 - 414 T + p^{3} T^{2} \) |
| 59 | \( 1 - 684 T + p^{3} T^{2} \) |
| 61 | \( 1 - 422 T + p^{3} T^{2} \) |
| 67 | \( 1 + 332 T + p^{3} T^{2} \) |
| 71 | \( 1 - 360 T + p^{3} T^{2} \) |
| 73 | \( 1 - 26 T + p^{3} T^{2} \) |
| 79 | \( 1 + 512 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1188 T + p^{3} T^{2} \) |
| 89 | \( 1 + 630 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1054 T + p^{3} T^{2} \) |
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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
−14.91136380842890822491522520847, −13.19912092156415129323863478153, −12.07806397603826663674915771745, −11.28054822259888086607869188422, −9.948507386257219990476757146924, −8.122343528408522207512179481216, −7.12426016835129266114591665157, −5.22660891485609789833003404037, −3.56656900484746205260423419053, 0,
3.56656900484746205260423419053, 5.22660891485609789833003404037, 7.12426016835129266114591665157, 8.122343528408522207512179481216, 9.948507386257219990476757146924, 11.28054822259888086607869188422, 12.07806397603826663674915771745, 13.19912092156415129323863478153, 14.91136380842890822491522520847