L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 2.23·11-s − 4.23·13-s − 14-s + 16-s + 0.381·17-s − 2.38·19-s + 2.23·22-s + 6.23·23-s + 4.23·26-s + 28-s + 3.47·29-s − 10.8·31-s − 32-s − 0.381·34-s + 6.47·37-s + 2.38·38-s + 10.0·41-s + 5.70·43-s − 2.23·44-s − 6.23·46-s + 4.38·47-s − 6·49-s − 4.23·52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.377·7-s − 0.353·8-s − 0.674·11-s − 1.17·13-s − 0.267·14-s + 0.250·16-s + 0.0926·17-s − 0.546·19-s + 0.476·22-s + 1.30·23-s + 0.830·26-s + 0.188·28-s + 0.644·29-s − 1.94·31-s − 0.176·32-s − 0.0655·34-s + 1.06·37-s + 0.386·38-s + 1.57·41-s + 0.870·43-s − 0.337·44-s − 0.919·46-s + 0.639·47-s − 0.857·49-s − 0.587·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 + 2.23T + 11T^{2} \) |
| 13 | \( 1 + 4.23T + 13T^{2} \) |
| 17 | \( 1 - 0.381T + 17T^{2} \) |
| 19 | \( 1 + 2.38T + 19T^{2} \) |
| 23 | \( 1 - 6.23T + 23T^{2} \) |
| 29 | \( 1 - 3.47T + 29T^{2} \) |
| 31 | \( 1 + 10.8T + 31T^{2} \) |
| 37 | \( 1 - 6.47T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 - 5.70T + 43T^{2} \) |
| 47 | \( 1 - 4.38T + 47T^{2} \) |
| 53 | \( 1 + 3.61T + 53T^{2} \) |
| 59 | \( 1 - 6.23T + 59T^{2} \) |
| 61 | \( 1 + 1.14T + 61T^{2} \) |
| 67 | \( 1 - 3.47T + 67T^{2} \) |
| 71 | \( 1 + 6.76T + 71T^{2} \) |
| 73 | \( 1 + 6.09T + 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 + 4.61T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−7.53070432643758528325066747309, −7.28303404884065924206891733140, −6.33087661176485900960600102145, −5.49506223262656559495373445011, −4.87426984952662239625132973165, −4.01247558514313038505325664696, −2.81713160495213403116785506831, −2.33287161423454338634488463996, −1.18346086040973081567699611629, 0,
1.18346086040973081567699611629, 2.33287161423454338634488463996, 2.81713160495213403116785506831, 4.01247558514313038505325664696, 4.87426984952662239625132973165, 5.49506223262656559495373445011, 6.33087661176485900960600102145, 7.28303404884065924206891733140, 7.53070432643758528325066747309