| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s − 4·13-s − 15-s − 3·17-s − 4·19-s − 21-s − 23-s + 25-s + 27-s − 3·29-s − 7·31-s + 35-s + 11·37-s − 4·39-s − 9·41-s − 4·43-s − 45-s + 6·47-s − 6·49-s − 3·51-s + 9·53-s − 4·57-s + 3·59-s − 10·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.10·13-s − 0.258·15-s − 0.727·17-s − 0.917·19-s − 0.218·21-s − 0.208·23-s + 1/5·25-s + 0.192·27-s − 0.557·29-s − 1.25·31-s + 0.169·35-s + 1.80·37-s − 0.640·39-s − 1.40·41-s − 0.609·43-s − 0.149·45-s + 0.875·47-s − 6/7·49-s − 0.420·51-s + 1.23·53-s − 0.529·57-s + 0.390·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−9.136723042105678626497401946106, −8.426712326417971032599906101216, −7.52652311249344567844634763071, −6.93543506214015159115646955058, −5.92285958674440455094839040519, −4.74028235498131449962795690414, −3.99337842949862406268567434883, −2.93143144537160068866262117724, −1.95878492833586522695243535909, 0,
1.95878492833586522695243535909, 2.93143144537160068866262117724, 3.99337842949862406268567434883, 4.74028235498131449962795690414, 5.92285958674440455094839040519, 6.93543506214015159115646955058, 7.52652311249344567844634763071, 8.426712326417971032599906101216, 9.136723042105678626497401946106
