| L(s) = 1 | + 2·3-s + 5-s − 7-s + 9-s − 4·13-s + 2·15-s − 4·19-s − 2·21-s + 25-s − 4·27-s − 6·29-s − 10·31-s − 35-s − 2·37-s − 8·39-s + 12·41-s − 4·43-s + 45-s + 6·47-s + 49-s + 6·53-s − 8·57-s + 6·59-s − 4·61-s − 63-s − 4·65-s + 4·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.10·13-s + 0.516·15-s − 0.917·19-s − 0.436·21-s + 1/5·25-s − 0.769·27-s − 1.11·29-s − 1.79·31-s − 0.169·35-s − 0.328·37-s − 1.28·39-s + 1.87·41-s − 0.609·43-s + 0.149·45-s + 0.875·47-s + 1/7·49-s + 0.824·53-s − 1.05·57-s + 0.781·59-s − 0.512·61-s − 0.125·63-s − 0.496·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 271040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 271040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.06587855060061, −12.67967093927241, −12.29372561858954, −11.62983150025042, −11.08790875038924, −10.64147782772110, −10.17322213156513, −9.545506003449938, −9.295672033966708, −8.979137966182096, −8.426476900105597, −7.875537235700697, −7.344766747959962, −7.166398214427624, −6.402700823607184, −5.883407316124669, −5.395404495684616, −4.905145953384325, −4.105157601852693, −3.758080345608300, −3.245109120845755, −2.504041849134403, −2.190961485044660, −1.838082083930258, −0.7600723272601106, 0,
0.7600723272601106, 1.838082083930258, 2.190961485044660, 2.504041849134403, 3.245109120845755, 3.758080345608300, 4.105157601852693, 4.905145953384325, 5.395404495684616, 5.883407316124669, 6.402700823607184, 7.166398214427624, 7.344766747959962, 7.875537235700697, 8.426476900105597, 8.979137966182096, 9.295672033966708, 9.545506003449938, 10.17322213156513, 10.64147782772110, 11.08790875038924, 11.62983150025042, 12.29372561858954, 12.67967093927241, 13.06587855060061
