L(s) = 1 | − 3-s + 1.52·5-s − 0.428·7-s + 9-s + 1.35·11-s + 5.07·13-s − 1.52·15-s + 3.18·17-s + 2.82·19-s + 0.428·21-s − 7.39·23-s − 2.66·25-s − 27-s − 1.58·29-s − 9.02·31-s − 1.35·33-s − 0.654·35-s − 4.97·37-s − 5.07·39-s − 9.59·41-s − 4.94·43-s + 1.52·45-s − 10.0·47-s − 6.81·49-s − 3.18·51-s + 9.53·53-s + 2.06·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.683·5-s − 0.161·7-s + 0.333·9-s + 0.407·11-s + 1.40·13-s − 0.394·15-s + 0.772·17-s + 0.647·19-s + 0.0935·21-s − 1.54·23-s − 0.533·25-s − 0.192·27-s − 0.293·29-s − 1.62·31-s − 0.235·33-s − 0.110·35-s − 0.818·37-s − 0.812·39-s − 1.49·41-s − 0.753·43-s + 0.227·45-s − 1.46·47-s − 0.973·49-s − 0.445·51-s + 1.31·53-s + 0.278·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 - T \) |
good | 5 | \( 1 - 1.52T + 5T^{2} \) |
| 7 | \( 1 + 0.428T + 7T^{2} \) |
| 11 | \( 1 - 1.35T + 11T^{2} \) |
| 13 | \( 1 - 5.07T + 13T^{2} \) |
| 17 | \( 1 - 3.18T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 7.39T + 23T^{2} \) |
| 29 | \( 1 + 1.58T + 29T^{2} \) |
| 31 | \( 1 + 9.02T + 31T^{2} \) |
| 37 | \( 1 + 4.97T + 37T^{2} \) |
| 41 | \( 1 + 9.59T + 41T^{2} \) |
| 43 | \( 1 + 4.94T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 - 9.53T + 53T^{2} \) |
| 59 | \( 1 + 7.81T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 - 6.88T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 - 10.8T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 + 8.75T + 89T^{2} \) |
| 97 | \( 1 - 2.00T + 97T^{2} \) |
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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
−7.42102825807409416078194480016, −6.61523249922371828100737047940, −6.01076377920958179342643120008, −5.60320104794135993152096505221, −4.83145886760288586353453715739, −3.67338264068992134898912763039, −3.42753495673263410306517172570, −1.86579598407258331133129957913, −1.44363514544066133930306265324, 0,
1.44363514544066133930306265324, 1.86579598407258331133129957913, 3.42753495673263410306517172570, 3.67338264068992134898912763039, 4.83145886760288586353453715739, 5.60320104794135993152096505221, 6.01076377920958179342643120008, 6.61523249922371828100737047940, 7.42102825807409416078194480016