L(s) = 1 | + 2-s − 3.09·3-s + 4-s + 3.37·5-s − 3.09·6-s + 1.17·7-s + 8-s + 6.57·9-s + 3.37·10-s − 1.16·11-s − 3.09·12-s + 7.15·13-s + 1.17·14-s − 10.4·15-s + 16-s + 5.64·17-s + 6.57·18-s + 0.667·19-s + 3.37·20-s − 3.62·21-s − 1.16·22-s + 2.74·23-s − 3.09·24-s + 6.37·25-s + 7.15·26-s − 11.0·27-s + 1.17·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.78·3-s + 0.5·4-s + 1.50·5-s − 1.26·6-s + 0.442·7-s + 0.353·8-s + 2.19·9-s + 1.06·10-s − 0.350·11-s − 0.893·12-s + 1.98·13-s + 0.313·14-s − 2.69·15-s + 0.250·16-s + 1.36·17-s + 1.55·18-s + 0.153·19-s + 0.754·20-s − 0.791·21-s − 0.247·22-s + 0.572·23-s − 0.631·24-s + 1.27·25-s + 1.40·26-s − 2.13·27-s + 0.221·28-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
\(L(1)\) |
\(\approx\) |
\(2.920647911\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.920647911\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.09T + 3T^{2} \) |
| 5 | \( 1 - 3.37T + 5T^{2} \) |
| 7 | \( 1 - 1.17T + 7T^{2} \) |
| 11 | \( 1 + 1.16T + 11T^{2} \) |
| 13 | \( 1 - 7.15T + 13T^{2} \) |
| 17 | \( 1 - 5.64T + 17T^{2} \) |
| 19 | \( 1 - 0.667T + 19T^{2} \) |
| 23 | \( 1 - 2.74T + 23T^{2} \) |
| 29 | \( 1 + 1.38T + 29T^{2} \) |
| 31 | \( 1 + 0.606T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 + 2.86T + 41T^{2} \) |
| 43 | \( 1 - 2.19T + 43T^{2} \) |
| 47 | \( 1 + 3.44T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 - 8.31T + 59T^{2} \) |
| 61 | \( 1 + 8.17T + 61T^{2} \) |
| 67 | \( 1 + 1.41T + 67T^{2} \) |
| 71 | \( 1 - 2.93T + 71T^{2} \) |
| 73 | \( 1 - 0.186T + 73T^{2} \) |
| 79 | \( 1 + 7.87T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 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
−8.326712600056133777348060830106, −7.40246246461890388844282253017, −6.40417168382146206835704189188, −6.10263678369837563119243180439, −5.52118521976523159762291314521, −5.06668761594840357556989383496, −4.13848985847576199758488560748, −3.03646008814056027570507506730, −1.59142187666833574236832300147, −1.13432616323081642396833744673,
1.13432616323081642396833744673, 1.59142187666833574236832300147, 3.03646008814056027570507506730, 4.13848985847576199758488560748, 5.06668761594840357556989383496, 5.52118521976523159762291314521, 6.10263678369837563119243180439, 6.40417168382146206835704189188, 7.40246246461890388844282253017, 8.326712600056133777348060830106