| L(s) = 1 | + 0.167·3-s − 3.79·5-s − 7-s − 2.97·9-s + 2.86·11-s − 0.635·15-s + 1.96·17-s + 7.08·19-s − 0.167·21-s − 5.54·23-s + 9.40·25-s − 1.00·27-s − 6.34·29-s + 8.34·31-s + 0.479·33-s + 3.79·35-s + 9.56·37-s − 8.01·41-s + 3.12·43-s + 11.2·45-s − 5.13·47-s + 49-s + 0.328·51-s + 0.726·53-s − 10.8·55-s + 1.18·57-s + 3.32·59-s + ⋯ |
| L(s) = 1 | + 0.0966·3-s − 1.69·5-s − 0.377·7-s − 0.990·9-s + 0.863·11-s − 0.164·15-s + 0.476·17-s + 1.62·19-s − 0.0365·21-s − 1.15·23-s + 1.88·25-s − 0.192·27-s − 1.17·29-s + 1.49·31-s + 0.0834·33-s + 0.641·35-s + 1.57·37-s − 1.25·41-s + 0.476·43-s + 1.68·45-s − 0.748·47-s + 0.142·49-s + 0.0460·51-s + 0.0997·53-s − 1.46·55-s + 0.157·57-s + 0.432·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4732 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4732 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 0.167T + 3T^{2} \) |
| 5 | \( 1 + 3.79T + 5T^{2} \) |
| 11 | \( 1 - 2.86T + 11T^{2} \) |
| 17 | \( 1 - 1.96T + 17T^{2} \) |
| 19 | \( 1 - 7.08T + 19T^{2} \) |
| 23 | \( 1 + 5.54T + 23T^{2} \) |
| 29 | \( 1 + 6.34T + 29T^{2} \) |
| 31 | \( 1 - 8.34T + 31T^{2} \) |
| 37 | \( 1 - 9.56T + 37T^{2} \) |
| 41 | \( 1 + 8.01T + 41T^{2} \) |
| 43 | \( 1 - 3.12T + 43T^{2} \) |
| 47 | \( 1 + 5.13T + 47T^{2} \) |
| 53 | \( 1 - 0.726T + 53T^{2} \) |
| 59 | \( 1 - 3.32T + 59T^{2} \) |
| 61 | \( 1 + 14.8T + 61T^{2} \) |
| 67 | \( 1 - 14.6T + 67T^{2} \) |
| 71 | \( 1 + 2.70T + 71T^{2} \) |
| 73 | \( 1 + 4.23T + 73T^{2} \) |
| 79 | \( 1 + 2.65T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 + 8.85T + 89T^{2} \) |
| 97 | \( 1 + 4.55T + 97T^{2} \) |
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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
−7.958733391926956097378525364844, −7.40869244487193310643713478734, −6.53726773308099305724355325067, −5.79256754604070019899468483674, −4.87294923426645210149408240002, −3.94721556506543954386653799468, −3.46322602143499718379446960527, −2.71815235475963988392048400142, −1.12107344993391167601854630677, 0,
1.12107344993391167601854630677, 2.71815235475963988392048400142, 3.46322602143499718379446960527, 3.94721556506543954386653799468, 4.87294923426645210149408240002, 5.79256754604070019899468483674, 6.53726773308099305724355325067, 7.40869244487193310643713478734, 7.958733391926956097378525364844
