| L(s) = 1 | + 8·3-s − 4·7-s + 37·9-s + 12·11-s − 58·13-s − 66·17-s − 100·19-s − 32·21-s + 132·23-s + 80·27-s + 90·29-s − 152·31-s + 96·33-s − 34·37-s − 464·39-s − 438·41-s − 32·43-s − 204·47-s − 327·49-s − 528·51-s + 222·53-s − 800·57-s + 420·59-s − 902·61-s − 148·63-s + 1.02e3·67-s + 1.05e3·69-s + ⋯ |
| L(s) = 1 | + 1.53·3-s − 0.215·7-s + 1.37·9-s + 0.328·11-s − 1.23·13-s − 0.941·17-s − 1.20·19-s − 0.332·21-s + 1.19·23-s + 0.570·27-s + 0.576·29-s − 0.880·31-s + 0.506·33-s − 0.151·37-s − 1.90·39-s − 1.66·41-s − 0.113·43-s − 0.633·47-s − 0.953·49-s − 1.44·51-s + 0.575·53-s − 1.85·57-s + 0.926·59-s − 1.89·61-s − 0.295·63-s + 1.86·67-s + 1.84·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 8 T + p^{3} T^{2} \) |
| 7 | \( 1 + 4 T + p^{3} T^{2} \) |
| 11 | \( 1 - 12 T + p^{3} T^{2} \) |
| 13 | \( 1 + 58 T + p^{3} T^{2} \) |
| 17 | \( 1 + 66 T + p^{3} T^{2} \) |
| 19 | \( 1 + 100 T + p^{3} T^{2} \) |
| 23 | \( 1 - 132 T + p^{3} T^{2} \) |
| 29 | \( 1 - 90 T + p^{3} T^{2} \) |
| 31 | \( 1 + 152 T + p^{3} T^{2} \) |
| 37 | \( 1 + 34 T + p^{3} T^{2} \) |
| 41 | \( 1 + 438 T + p^{3} T^{2} \) |
| 43 | \( 1 + 32 T + p^{3} T^{2} \) |
| 47 | \( 1 + 204 T + p^{3} T^{2} \) |
| 53 | \( 1 - 222 T + p^{3} T^{2} \) |
| 59 | \( 1 - 420 T + p^{3} T^{2} \) |
| 61 | \( 1 + 902 T + p^{3} T^{2} \) |
| 67 | \( 1 - 1024 T + p^{3} T^{2} \) |
| 71 | \( 1 + 432 T + p^{3} T^{2} \) |
| 73 | \( 1 + 362 T + p^{3} T^{2} \) |
| 79 | \( 1 - 160 T + p^{3} T^{2} \) |
| 83 | \( 1 + 72 T + p^{3} T^{2} \) |
| 89 | \( 1 - 810 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1106 T + p^{3} T^{2} \) |
| 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
−8.713119708209427617064257199100, −8.032074868878772243840447054463, −7.07589995542523419959382738100, −6.58708086628972957656195052676, −5.10462761520489991869380359426, −4.28783118974605022607906101694, −3.33017316878570444266183831136, −2.52221873360286908650544342589, −1.72483963580542929396667597854, 0,
1.72483963580542929396667597854, 2.52221873360286908650544342589, 3.33017316878570444266183831136, 4.28783118974605022607906101694, 5.10462761520489991869380359426, 6.58708086628972957656195052676, 7.07589995542523419959382738100, 8.032074868878772243840447054463, 8.713119708209427617064257199100
