L(s) = 1 | + 0.00508·3-s − 3.22·5-s − 4.03·7-s − 2.99·9-s + 0.837·11-s + 2.15·13-s − 0.0164·15-s − 5.70·17-s − 3.20·19-s − 0.0205·21-s − 7.31·23-s + 5.40·25-s − 0.0305·27-s − 10.2·29-s − 2.92·31-s + 0.00426·33-s + 13.0·35-s + 1.03·37-s + 0.0109·39-s − 6.57·41-s + 8.92·43-s + 9.67·45-s + 0.200·47-s + 9.30·49-s − 0.0290·51-s − 4.20·53-s − 2.70·55-s + ⋯ |
L(s) = 1 | + 0.00293·3-s − 1.44·5-s − 1.52·7-s − 0.999·9-s + 0.252·11-s + 0.597·13-s − 0.00423·15-s − 1.38·17-s − 0.735·19-s − 0.00448·21-s − 1.52·23-s + 1.08·25-s − 0.00587·27-s − 1.89·29-s − 0.525·31-s + 0.000741·33-s + 2.20·35-s + 0.170·37-s + 0.00175·39-s − 1.02·41-s + 1.36·43-s + 1.44·45-s + 0.0292·47-s + 1.32·49-s − 0.00406·51-s − 0.577·53-s − 0.364·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.09893459332\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09893459332\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
good | 3 | \( 1 - 0.00508T + 3T^{2} \) |
| 5 | \( 1 + 3.22T + 5T^{2} \) |
| 7 | \( 1 + 4.03T + 7T^{2} \) |
| 11 | \( 1 - 0.837T + 11T^{2} \) |
| 13 | \( 1 - 2.15T + 13T^{2} \) |
| 17 | \( 1 + 5.70T + 17T^{2} \) |
| 19 | \( 1 + 3.20T + 19T^{2} \) |
| 23 | \( 1 + 7.31T + 23T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 + 2.92T + 31T^{2} \) |
| 37 | \( 1 - 1.03T + 37T^{2} \) |
| 41 | \( 1 + 6.57T + 41T^{2} \) |
| 43 | \( 1 - 8.92T + 43T^{2} \) |
| 47 | \( 1 - 0.200T + 47T^{2} \) |
| 53 | \( 1 + 4.20T + 53T^{2} \) |
| 59 | \( 1 - 4.19T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 - 5.06T + 67T^{2} \) |
| 71 | \( 1 - 6.00T + 71T^{2} \) |
| 73 | \( 1 + 3.36T + 73T^{2} \) |
| 79 | \( 1 + 6.23T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 + 7.88T + 89T^{2} \) |
| 97 | \( 1 - 4.91T + 97T^{2} \) |
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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
−8.519077399438792171796073793942, −7.73515345260515228600546736569, −6.97125330448816722976523692667, −6.24764857475565236905640541014, −5.71238465185635171779664628973, −4.32803781282621288337944353056, −3.81681942806280160124919532926, −3.19444218046377932613847460665, −2.12932585250450049667872892084, −0.16582806354601031248151206204,
0.16582806354601031248151206204, 2.12932585250450049667872892084, 3.19444218046377932613847460665, 3.81681942806280160124919532926, 4.32803781282621288337944353056, 5.71238465185635171779664628973, 6.24764857475565236905640541014, 6.97125330448816722976523692667, 7.73515345260515228600546736569, 8.519077399438792171796073793942