| L(s) = 1 | − 2.58·3-s − 2.81·7-s + 3.69·9-s − 1.58·11-s + 0.888·13-s − 3.98·17-s − 19-s + 7.27·21-s − 3.30·23-s − 1.81·27-s + 6.98·29-s + 4.51·31-s + 4.11·33-s − 2.41·37-s − 2.30·39-s + 5.09·41-s − 8.17·43-s + 9.08·47-s + 0.901·49-s + 10.3·51-s + 7.79·53-s + 2.58·57-s + 2.22·59-s − 9.90·61-s − 10.3·63-s + 7.56·67-s + 8.54·69-s + ⋯ |
| L(s) = 1 | − 1.49·3-s − 1.06·7-s + 1.23·9-s − 0.478·11-s + 0.246·13-s − 0.967·17-s − 0.229·19-s + 1.58·21-s − 0.688·23-s − 0.348·27-s + 1.29·29-s + 0.810·31-s + 0.715·33-s − 0.396·37-s − 0.368·39-s + 0.796·41-s − 1.24·43-s + 1.32·47-s + 0.128·49-s + 1.44·51-s + 1.07·53-s + 0.342·57-s + 0.289·59-s − 1.26·61-s − 1.31·63-s + 0.924·67-s + 1.02·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 2.58T + 3T^{2} \) |
| 7 | \( 1 + 2.81T + 7T^{2} \) |
| 11 | \( 1 + 1.58T + 11T^{2} \) |
| 13 | \( 1 - 0.888T + 13T^{2} \) |
| 17 | \( 1 + 3.98T + 17T^{2} \) |
| 23 | \( 1 + 3.30T + 23T^{2} \) |
| 29 | \( 1 - 6.98T + 29T^{2} \) |
| 31 | \( 1 - 4.51T + 31T^{2} \) |
| 37 | \( 1 + 2.41T + 37T^{2} \) |
| 41 | \( 1 - 5.09T + 41T^{2} \) |
| 43 | \( 1 + 8.17T + 43T^{2} \) |
| 47 | \( 1 - 9.08T + 47T^{2} \) |
| 53 | \( 1 - 7.79T + 53T^{2} \) |
| 59 | \( 1 - 2.22T + 59T^{2} \) |
| 61 | \( 1 + 9.90T + 61T^{2} \) |
| 67 | \( 1 - 7.56T + 67T^{2} \) |
| 71 | \( 1 - 0.777T + 71T^{2} \) |
| 73 | \( 1 + 0.876T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 + 5.27T + 89T^{2} \) |
| 97 | \( 1 - 7.24T + 97T^{2} \) |
| 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
−7.22243732903129305053860312198, −6.61031754064496319688759869865, −6.19413744345440273621967257929, −5.58426795495178321528547323107, −4.75570186883433700774176883568, −4.17144770869273267004264896295, −3.14009311468602493960715291519, −2.22591496639480145714983371725, −0.873301511806359583044717761745, 0,
0.873301511806359583044717761745, 2.22591496639480145714983371725, 3.14009311468602493960715291519, 4.17144770869273267004264896295, 4.75570186883433700774176883568, 5.58426795495178321528547323107, 6.19413744345440273621967257929, 6.61031754064496319688759869865, 7.22243732903129305053860312198
