| L(s) = 1 | + 2.32·3-s + 0.425·5-s − 5.05·7-s + 2.40·9-s + 4.95·11-s − 3.33·13-s + 0.989·15-s − 0.504·17-s + 0.518·19-s − 11.7·21-s + 5.66·23-s − 4.81·25-s − 1.38·27-s + 2.31·29-s − 8.44·31-s + 11.5·33-s − 2.15·35-s + 0.409·37-s − 7.74·39-s + 0.517·41-s − 5.57·43-s + 1.02·45-s − 12.6·47-s + 18.5·49-s − 1.17·51-s − 8.34·53-s + 2.10·55-s + ⋯ |
| L(s) = 1 | + 1.34·3-s + 0.190·5-s − 1.90·7-s + 0.801·9-s + 1.49·11-s − 0.924·13-s + 0.255·15-s − 0.122·17-s + 0.118·19-s − 2.56·21-s + 1.18·23-s − 0.963·25-s − 0.266·27-s + 0.430·29-s − 1.51·31-s + 2.00·33-s − 0.363·35-s + 0.0673·37-s − 1.24·39-s + 0.0807·41-s − 0.849·43-s + 0.152·45-s − 1.85·47-s + 2.64·49-s − 0.164·51-s − 1.14·53-s + 0.284·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 - 2.32T + 3T^{2} \) |
| 5 | \( 1 - 0.425T + 5T^{2} \) |
| 7 | \( 1 + 5.05T + 7T^{2} \) |
| 11 | \( 1 - 4.95T + 11T^{2} \) |
| 13 | \( 1 + 3.33T + 13T^{2} \) |
| 17 | \( 1 + 0.504T + 17T^{2} \) |
| 19 | \( 1 - 0.518T + 19T^{2} \) |
| 23 | \( 1 - 5.66T + 23T^{2} \) |
| 29 | \( 1 - 2.31T + 29T^{2} \) |
| 31 | \( 1 + 8.44T + 31T^{2} \) |
| 37 | \( 1 - 0.409T + 37T^{2} \) |
| 41 | \( 1 - 0.517T + 41T^{2} \) |
| 43 | \( 1 + 5.57T + 43T^{2} \) |
| 47 | \( 1 + 12.6T + 47T^{2} \) |
| 53 | \( 1 + 8.34T + 53T^{2} \) |
| 59 | \( 1 + 1.25T + 59T^{2} \) |
| 61 | \( 1 + 7.52T + 61T^{2} \) |
| 67 | \( 1 - 6.33T + 67T^{2} \) |
| 71 | \( 1 + 1.91T + 71T^{2} \) |
| 73 | \( 1 + 1.06T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 + 17.5T + 83T^{2} \) |
| 89 | \( 1 - 6.66T + 89T^{2} \) |
| 97 | \( 1 - 0.181T + 97T^{2} \) |
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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
−7.70056141547212790392899861521, −6.96536733281352531715280569228, −6.55449757718815316832194958690, −5.74393639721742078812190982740, −4.62217513450058883107697755787, −3.58824901806427839713825686192, −3.35613697201954112346335254825, −2.52734050494132675890801965909, −1.56632717641029567224967169490, 0,
1.56632717641029567224967169490, 2.52734050494132675890801965909, 3.35613697201954112346335254825, 3.58824901806427839713825686192, 4.62217513450058883107697755787, 5.74393639721742078812190982740, 6.55449757718815316832194958690, 6.96536733281352531715280569228, 7.70056141547212790392899861521
