| L(s) = 1 | − 2.09·3-s − 2.89·5-s − 1.97·7-s + 1.40·9-s − 4.41·11-s + 5.29·13-s + 6.06·15-s + 4.12·17-s − 5.91·19-s + 4.14·21-s − 4.03·23-s + 3.35·25-s + 3.33·27-s + 2.04·29-s − 9.54·31-s + 9.26·33-s + 5.69·35-s + 3.08·37-s − 11.1·39-s + 4.34·41-s + 1.34·43-s − 4.07·45-s + 7.33·47-s − 3.11·49-s − 8.65·51-s + 7.15·53-s + 12.7·55-s + ⋯ |
| L(s) = 1 | − 1.21·3-s − 1.29·5-s − 0.745·7-s + 0.469·9-s − 1.33·11-s + 1.46·13-s + 1.56·15-s + 0.999·17-s − 1.35·19-s + 0.903·21-s − 0.842·23-s + 0.670·25-s + 0.642·27-s + 0.379·29-s − 1.71·31-s + 1.61·33-s + 0.963·35-s + 0.507·37-s − 1.77·39-s + 0.678·41-s + 0.205·43-s − 0.607·45-s + 1.07·47-s − 0.444·49-s − 1.21·51-s + 0.982·53-s + 1.71·55-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 | \( 1 + 2.09T + 3T^{2} \) |
| 5 | \( 1 + 2.89T + 5T^{2} \) |
| 7 | \( 1 + 1.97T + 7T^{2} \) |
| 11 | \( 1 + 4.41T + 11T^{2} \) |
| 13 | \( 1 - 5.29T + 13T^{2} \) |
| 17 | \( 1 - 4.12T + 17T^{2} \) |
| 19 | \( 1 + 5.91T + 19T^{2} \) |
| 23 | \( 1 + 4.03T + 23T^{2} \) |
| 29 | \( 1 - 2.04T + 29T^{2} \) |
| 31 | \( 1 + 9.54T + 31T^{2} \) |
| 37 | \( 1 - 3.08T + 37T^{2} \) |
| 41 | \( 1 - 4.34T + 41T^{2} \) |
| 43 | \( 1 - 1.34T + 43T^{2} \) |
| 47 | \( 1 - 7.33T + 47T^{2} \) |
| 53 | \( 1 - 7.15T + 53T^{2} \) |
| 59 | \( 1 - 7.09T + 59T^{2} \) |
| 61 | \( 1 + 2.47T + 61T^{2} \) |
| 67 | \( 1 - 15.5T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 - 8.28T + 73T^{2} \) |
| 79 | \( 1 - 7.28T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 + 5.72T + 89T^{2} \) |
| 97 | \( 1 + 3.73T + 97T^{2} \) |
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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
−7.75084480779333606549305878377, −6.96386771413067900267583938162, −6.11816541348087092713954129941, −5.75115724328329137078852205474, −4.92200926494484818923123581396, −3.93708397552830770428193812775, −3.54359533704467948071236504169, −2.37861301843431720340927600110, −0.807179065988432061005270006012, 0,
0.807179065988432061005270006012, 2.37861301843431720340927600110, 3.54359533704467948071236504169, 3.93708397552830770428193812775, 4.92200926494484818923123581396, 5.75115724328329137078852205474, 6.11816541348087092713954129941, 6.96386771413067900267583938162, 7.75084480779333606549305878377
