| L(s) = 1 | − 2.60·2-s − 1.96·3-s + 4.78·4-s + 5-s + 5.12·6-s + 7-s − 7.26·8-s + 0.865·9-s − 2.60·10-s + 0.861·11-s − 9.41·12-s + 1.22·13-s − 2.60·14-s − 1.96·15-s + 9.36·16-s + 6.95·17-s − 2.25·18-s + 3.11·19-s + 4.78·20-s − 1.96·21-s − 2.24·22-s + 5.50·23-s + 14.2·24-s + 25-s − 3.19·26-s + 4.19·27-s + 4.78·28-s + ⋯ |
| L(s) = 1 | − 1.84·2-s − 1.13·3-s + 2.39·4-s + 0.447·5-s + 2.09·6-s + 0.377·7-s − 2.56·8-s + 0.288·9-s − 0.823·10-s + 0.259·11-s − 2.71·12-s + 0.339·13-s − 0.696·14-s − 0.507·15-s + 2.34·16-s + 1.68·17-s − 0.531·18-s + 0.715·19-s + 1.07·20-s − 0.429·21-s − 0.478·22-s + 1.14·23-s + 2.91·24-s + 0.200·25-s − 0.625·26-s + 0.807·27-s + 0.905·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 + T \) |
| good | 2 | \( 1 + 2.60T + 2T^{2} \) |
| 3 | \( 1 + 1.96T + 3T^{2} \) |
| 11 | \( 1 - 0.861T + 11T^{2} \) |
| 13 | \( 1 - 1.22T + 13T^{2} \) |
| 17 | \( 1 - 6.95T + 17T^{2} \) |
| 19 | \( 1 - 3.11T + 19T^{2} \) |
| 23 | \( 1 - 5.50T + 23T^{2} \) |
| 29 | \( 1 + 7.82T + 29T^{2} \) |
| 31 | \( 1 + 7.87T + 31T^{2} \) |
| 37 | \( 1 - 1.73T + 37T^{2} \) |
| 41 | \( 1 + 4.72T + 41T^{2} \) |
| 43 | \( 1 + 8.26T + 43T^{2} \) |
| 47 | \( 1 + 0.377T + 47T^{2} \) |
| 53 | \( 1 + 0.141T + 53T^{2} \) |
| 59 | \( 1 - 4.69T + 59T^{2} \) |
| 61 | \( 1 - 2.58T + 61T^{2} \) |
| 67 | \( 1 + 0.162T + 67T^{2} \) |
| 71 | \( 1 - 2.32T + 71T^{2} \) |
| 73 | \( 1 - 4.98T + 73T^{2} \) |
| 79 | \( 1 + 8.99T + 79T^{2} \) |
| 83 | \( 1 + 9.60T + 83T^{2} \) |
| 89 | \( 1 + 7.56T + 89T^{2} \) |
| 97 | \( 1 + 9.53T + 97T^{2} \) |
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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
−7.43516150780530544340906145347, −7.04746323659386765009826671112, −6.27458129930807717313234425180, −5.46152846384245769860928008040, −5.26996445434841196186211440279, −3.66274620219958622000596519552, −2.79338845187953646476632543673, −1.55662311045458278341178892276, −1.14687369635532317186333031957, 0,
1.14687369635532317186333031957, 1.55662311045458278341178892276, 2.79338845187953646476632543673, 3.66274620219958622000596519552, 5.26996445434841196186211440279, 5.46152846384245769860928008040, 6.27458129930807717313234425180, 7.04746323659386765009826671112, 7.43516150780530544340906145347
