L(s) = 1 | + 0.643·3-s + 1.53·5-s − 1.96·7-s − 2.58·9-s − 3.95·11-s + 1.80·13-s + 0.984·15-s + 5.54·17-s + 8.43·19-s − 1.26·21-s + 0.611·23-s − 2.65·25-s − 3.59·27-s − 4.59·29-s − 8.83·31-s − 2.54·33-s − 3.00·35-s − 3.02·37-s + 1.15·39-s − 11.2·41-s + 6.62·43-s − 3.95·45-s + 6.14·47-s − 3.15·49-s + 3.56·51-s + 1.80·53-s − 6.05·55-s + ⋯ |
L(s) = 1 | + 0.371·3-s + 0.684·5-s − 0.740·7-s − 0.862·9-s − 1.19·11-s + 0.499·13-s + 0.254·15-s + 1.34·17-s + 1.93·19-s − 0.275·21-s + 0.127·23-s − 0.531·25-s − 0.691·27-s − 0.852·29-s − 1.58·31-s − 0.442·33-s − 0.507·35-s − 0.497·37-s + 0.185·39-s − 1.75·41-s + 1.01·43-s − 0.590·45-s + 0.895·47-s − 0.450·49-s + 0.499·51-s + 0.247·53-s − 0.815·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 751 | \( 1 - T \) |
good | 3 | \( 1 - 0.643T + 3T^{2} \) |
| 5 | \( 1 - 1.53T + 5T^{2} \) |
| 7 | \( 1 + 1.96T + 7T^{2} \) |
| 11 | \( 1 + 3.95T + 11T^{2} \) |
| 13 | \( 1 - 1.80T + 13T^{2} \) |
| 17 | \( 1 - 5.54T + 17T^{2} \) |
| 19 | \( 1 - 8.43T + 19T^{2} \) |
| 23 | \( 1 - 0.611T + 23T^{2} \) |
| 29 | \( 1 + 4.59T + 29T^{2} \) |
| 31 | \( 1 + 8.83T + 31T^{2} \) |
| 37 | \( 1 + 3.02T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 - 6.62T + 43T^{2} \) |
| 47 | \( 1 - 6.14T + 47T^{2} \) |
| 53 | \( 1 - 1.80T + 53T^{2} \) |
| 59 | \( 1 - 9.33T + 59T^{2} \) |
| 61 | \( 1 + 5.06T + 61T^{2} \) |
| 67 | \( 1 - 8.58T + 67T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 73 | \( 1 + 16.6T + 73T^{2} \) |
| 79 | \( 1 - 8.34T + 79T^{2} \) |
| 83 | \( 1 + 3.76T + 83T^{2} \) |
| 89 | \( 1 + 4.69T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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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
−7.55913325511285898129278091328, −7.31415482928490696878108625454, −6.05074827730179609226382461716, −5.51153270364581446924002762117, −5.27166304938344097706562497351, −3.65062657974430001475879193243, −3.25066458481247104338205575937, −2.47848555303697386399815282653, −1.41032713393010735930476735304, 0,
1.41032713393010735930476735304, 2.47848555303697386399815282653, 3.25066458481247104338205575937, 3.65062657974430001475879193243, 5.27166304938344097706562497351, 5.51153270364581446924002762117, 6.05074827730179609226382461716, 7.31415482928490696878108625454, 7.55913325511285898129278091328