L(s) = 1 | − 0.859·3-s + 2.49·5-s − 3.53·7-s − 2.26·9-s + 1.54·11-s + 3.17·13-s − 2.14·15-s − 17-s + 0.523·19-s + 3.03·21-s − 5.23·23-s + 1.24·25-s + 4.52·27-s + 4.96·29-s + 3.14·31-s − 1.32·33-s − 8.82·35-s − 7.80·37-s − 2.72·39-s − 1.58·41-s − 1.96·43-s − 5.64·45-s + 2.57·47-s + 5.48·49-s + 0.859·51-s + 2.58·53-s + 3.85·55-s + ⋯ |
L(s) = 1 | − 0.496·3-s + 1.11·5-s − 1.33·7-s − 0.753·9-s + 0.464·11-s + 0.880·13-s − 0.554·15-s − 0.242·17-s + 0.120·19-s + 0.663·21-s − 1.09·23-s + 0.248·25-s + 0.870·27-s + 0.921·29-s + 0.564·31-s − 0.230·33-s − 1.49·35-s − 1.28·37-s − 0.436·39-s − 0.248·41-s − 0.299·43-s − 0.841·45-s + 0.376·47-s + 0.784·49-s + 0.120·51-s + 0.355·53-s + 0.519·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
good | 3 | \( 1 + 0.859T + 3T^{2} \) |
| 5 | \( 1 - 2.49T + 5T^{2} \) |
| 7 | \( 1 + 3.53T + 7T^{2} \) |
| 11 | \( 1 - 1.54T + 11T^{2} \) |
| 13 | \( 1 - 3.17T + 13T^{2} \) |
| 19 | \( 1 - 0.523T + 19T^{2} \) |
| 23 | \( 1 + 5.23T + 23T^{2} \) |
| 29 | \( 1 - 4.96T + 29T^{2} \) |
| 31 | \( 1 - 3.14T + 31T^{2} \) |
| 37 | \( 1 + 7.80T + 37T^{2} \) |
| 41 | \( 1 + 1.58T + 41T^{2} \) |
| 43 | \( 1 + 1.96T + 43T^{2} \) |
| 47 | \( 1 - 2.57T + 47T^{2} \) |
| 53 | \( 1 - 2.58T + 53T^{2} \) |
| 61 | \( 1 - 6.54T + 61T^{2} \) |
| 67 | \( 1 + 5.68T + 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 + 1.37T + 73T^{2} \) |
| 79 | \( 1 + 6.64T + 79T^{2} \) |
| 83 | \( 1 + 16.8T + 83T^{2} \) |
| 89 | \( 1 - 2.73T + 89T^{2} \) |
| 97 | \( 1 - 3.59T + 97T^{2} \) |
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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
−8.331559340468110095993848868804, −6.99428823283412175178905440786, −6.41576472722736345329033419043, −5.95227349140632384642525898146, −5.42386655299874937769558624630, −4.24027292176833046285234201450, −3.31331383234736153623435954202, −2.53526196608386433408047714065, −1.37245905431549897903115807811, 0,
1.37245905431549897903115807811, 2.53526196608386433408047714065, 3.31331383234736153623435954202, 4.24027292176833046285234201450, 5.42386655299874937769558624630, 5.95227349140632384642525898146, 6.41576472722736345329033419043, 6.99428823283412175178905440786, 8.331559340468110095993848868804