L(s) = 1 | + 2.76·3-s − 0.761·7-s + 4.62·9-s + 0.864·11-s − 5.62·13-s + 3.62·17-s − 19-s − 2.10·21-s − 8.01·23-s + 4.49·27-s − 7.35·29-s − 8.11·31-s + 2.38·33-s − 0.476·37-s − 15.5·39-s − 2.65·41-s + 6.86·43-s + 1.25·47-s − 6.42·49-s + 10.0·51-s − 2.37·53-s − 2.76·57-s − 4.49·59-s − 10.8·61-s − 3.52·63-s + 1.03·67-s − 22.1·69-s + ⋯ |
L(s) = 1 | + 1.59·3-s − 0.287·7-s + 1.54·9-s + 0.260·11-s − 1.56·13-s + 0.879·17-s − 0.229·19-s − 0.458·21-s − 1.67·23-s + 0.864·27-s − 1.36·29-s − 1.45·31-s + 0.415·33-s − 0.0783·37-s − 2.48·39-s − 0.415·41-s + 1.04·43-s + 0.182·47-s − 0.917·49-s + 1.40·51-s − 0.326·53-s − 0.365·57-s − 0.584·59-s − 1.39·61-s − 0.443·63-s + 0.126·67-s − 2.66·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 - 2.76T + 3T^{2} \) |
| 7 | \( 1 + 0.761T + 7T^{2} \) |
| 11 | \( 1 - 0.864T + 11T^{2} \) |
| 13 | \( 1 + 5.62T + 13T^{2} \) |
| 17 | \( 1 - 3.62T + 17T^{2} \) |
| 23 | \( 1 + 8.01T + 23T^{2} \) |
| 29 | \( 1 + 7.35T + 29T^{2} \) |
| 31 | \( 1 + 8.11T + 31T^{2} \) |
| 37 | \( 1 + 0.476T + 37T^{2} \) |
| 41 | \( 1 + 2.65T + 41T^{2} \) |
| 43 | \( 1 - 6.86T + 43T^{2} \) |
| 47 | \( 1 - 1.25T + 47T^{2} \) |
| 53 | \( 1 + 2.37T + 53T^{2} \) |
| 59 | \( 1 + 4.49T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 - 1.03T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 + 16.4T + 73T^{2} \) |
| 79 | \( 1 - 12.5T + 79T^{2} \) |
| 83 | \( 1 + 0.270T + 83T^{2} \) |
| 89 | \( 1 - 0.387T + 89T^{2} \) |
| 97 | \( 1 - 8.50T + 97T^{2} \) |
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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
−7.62840360049169691250969163085, −7.23042360596903682474129864534, −6.19281825853353597927133504857, −5.40572497489090618617388704028, −4.45252891766792756518042616389, −3.72251025142018003712304971843, −3.15888418757372381564348366547, −2.24354585347192392172590967540, −1.72852559164119865953863530592, 0,
1.72852559164119865953863530592, 2.24354585347192392172590967540, 3.15888418757372381564348366547, 3.72251025142018003712304971843, 4.45252891766792756518042616389, 5.40572497489090618617388704028, 6.19281825853353597927133504857, 7.23042360596903682474129864534, 7.62840360049169691250969163085