L(s) = 1 | + 2-s − 0.366·3-s + 4-s + 0.145·5-s − 0.366·6-s + 3.13·7-s + 8-s − 2.86·9-s + 0.145·10-s − 2.53·11-s − 0.366·12-s − 2.06·13-s + 3.13·14-s − 0.0533·15-s + 16-s + 4.60·17-s − 2.86·18-s − 6.21·19-s + 0.145·20-s − 1.14·21-s − 2.53·22-s − 6.81·23-s − 0.366·24-s − 4.97·25-s − 2.06·26-s + 2.14·27-s + 3.13·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.211·3-s + 0.5·4-s + 0.0650·5-s − 0.149·6-s + 1.18·7-s + 0.353·8-s − 0.955·9-s + 0.0460·10-s − 0.765·11-s − 0.105·12-s − 0.573·13-s + 0.837·14-s − 0.0137·15-s + 0.250·16-s + 1.11·17-s − 0.675·18-s − 1.42·19-s + 0.0325·20-s − 0.250·21-s − 0.541·22-s − 1.42·23-s − 0.0747·24-s − 0.995·25-s − 0.405·26-s + 0.413·27-s + 0.592·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 2017 | \( 1 - T \) |
good | 3 | \( 1 + 0.366T + 3T^{2} \) |
| 5 | \( 1 - 0.145T + 5T^{2} \) |
| 7 | \( 1 - 3.13T + 7T^{2} \) |
| 11 | \( 1 + 2.53T + 11T^{2} \) |
| 13 | \( 1 + 2.06T + 13T^{2} \) |
| 17 | \( 1 - 4.60T + 17T^{2} \) |
| 19 | \( 1 + 6.21T + 19T^{2} \) |
| 23 | \( 1 + 6.81T + 23T^{2} \) |
| 29 | \( 1 + 4.38T + 29T^{2} \) |
| 31 | \( 1 - 1.20T + 31T^{2} \) |
| 37 | \( 1 + 2.43T + 37T^{2} \) |
| 41 | \( 1 - 3.27T + 41T^{2} \) |
| 43 | \( 1 + 4.27T + 43T^{2} \) |
| 47 | \( 1 - 9.97T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 + 9.09T + 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 + 2.84T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 6.04T + 73T^{2} \) |
| 79 | \( 1 + 2.04T + 79T^{2} \) |
| 83 | \( 1 + 4.46T + 83T^{2} \) |
| 89 | \( 1 + 2.54T + 89T^{2} \) |
| 97 | \( 1 - 16.3T + 97T^{2} \) |
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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
−7.84460916589820397350573242229, −7.61031850299862838983967119731, −6.24956709888014154312240146810, −5.79860841668703416695072622790, −5.07708516368346858469259754873, −4.43464982777019656642027902570, −3.48407402935522878723694045983, −2.43139026088455057842889095431, −1.75756557146202436324121925154, 0,
1.75756557146202436324121925154, 2.43139026088455057842889095431, 3.48407402935522878723694045983, 4.43464982777019656642027902570, 5.07708516368346858469259754873, 5.79860841668703416695072622790, 6.24956709888014154312240146810, 7.61031850299862838983967119731, 7.84460916589820397350573242229