L(s) = 1 | + 3-s + 0.0425·5-s − 1.43·7-s + 9-s + 1.32·11-s + 6.34·13-s + 0.0425·15-s − 3.59·17-s − 2.26·19-s − 1.43·21-s − 3.90·23-s − 4.99·25-s + 27-s − 8.64·29-s + 1.32·31-s + 1.32·33-s − 0.0610·35-s + 6.05·37-s + 6.34·39-s − 8.13·41-s + 5.53·43-s + 0.0425·45-s − 10.3·47-s − 4.93·49-s − 3.59·51-s − 7.21·53-s + 0.0564·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.0190·5-s − 0.542·7-s + 0.333·9-s + 0.400·11-s + 1.75·13-s + 0.0109·15-s − 0.871·17-s − 0.520·19-s − 0.313·21-s − 0.813·23-s − 0.999·25-s + 0.192·27-s − 1.60·29-s + 0.238·31-s + 0.231·33-s − 0.0103·35-s + 0.995·37-s + 1.01·39-s − 1.27·41-s + 0.844·43-s + 0.00633·45-s − 1.50·47-s − 0.705·49-s − 0.502·51-s − 0.990·53-s + 0.00761·55-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 | \( 1 - 0.0425T + 5T^{2} \) |
| 7 | \( 1 + 1.43T + 7T^{2} \) |
| 11 | \( 1 - 1.32T + 11T^{2} \) |
| 13 | \( 1 - 6.34T + 13T^{2} \) |
| 17 | \( 1 + 3.59T + 17T^{2} \) |
| 19 | \( 1 + 2.26T + 19T^{2} \) |
| 23 | \( 1 + 3.90T + 23T^{2} \) |
| 29 | \( 1 + 8.64T + 29T^{2} \) |
| 31 | \( 1 - 1.32T + 31T^{2} \) |
| 37 | \( 1 - 6.05T + 37T^{2} \) |
| 41 | \( 1 + 8.13T + 41T^{2} \) |
| 43 | \( 1 - 5.53T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 + 7.21T + 53T^{2} \) |
| 59 | \( 1 - 2.51T + 59T^{2} \) |
| 61 | \( 1 - 1.35T + 61T^{2} \) |
| 67 | \( 1 - 1.90T + 67T^{2} \) |
| 71 | \( 1 + 8.26T + 71T^{2} \) |
| 73 | \( 1 + 1.99T + 73T^{2} \) |
| 79 | \( 1 - 5.18T + 79T^{2} \) |
| 83 | \( 1 - 2.42T + 83T^{2} \) |
| 89 | \( 1 - 12.0T + 89T^{2} \) |
| 97 | \( 1 + 4.77T + 97T^{2} \) |
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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
−7.67454565231924281330330225079, −6.50792891633082654646783011929, −6.38666790933592726269453031605, −5.53590171527727277814307229785, −4.38797559332853565587864432259, −3.81969078519662491733522769508, −3.27379107741395648074016779125, −2.15149606833635619728267220300, −1.46182295332302988660850216447, 0,
1.46182295332302988660850216447, 2.15149606833635619728267220300, 3.27379107741395648074016779125, 3.81969078519662491733522769508, 4.38797559332853565587864432259, 5.53590171527727277814307229785, 6.38666790933592726269453031605, 6.50792891633082654646783011929, 7.67454565231924281330330225079