| L(s) = 1 | − 2·4-s − 4.76·7-s + 1.73·13-s + 4·16-s + 8.10·19-s − 5·25-s + 9.52·28-s − 10.3·31-s + 4.10·37-s + 12.1·43-s + 15.6·49-s − 3.46·52-s − 10.3·61-s − 8·64-s + 0.795·67-s + 13.8·73-s − 16.2·76-s + 15.0·79-s − 8.24·91-s + 0.101·97-s + 10·100-s − 20.2·103-s + 16.1·109-s − 19.0·112-s + ⋯ |
| L(s) = 1 | − 4-s − 1.79·7-s + 0.480·13-s + 16-s + 1.85·19-s − 25-s + 1.79·28-s − 1.86·31-s + 0.674·37-s + 1.84·43-s + 2.23·49-s − 0.480·52-s − 1.32·61-s − 64-s + 0.0971·67-s + 1.62·73-s − 1.85·76-s + 1.69·79-s − 0.864·91-s + 0.0103·97-s + 100-s − 1.99·103-s + 1.54·109-s − 1.79·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4761 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4761 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 + 4.76T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 1.73T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 8.10T + 19T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 - 4.10T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 12.1T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 - 0.795T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 0.101T + 97T^{2} \) |
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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
−7.79557568480264821856777293184, −7.36633506980232662161209317627, −6.30707578029302709754962316307, −5.76565435697176124308807435048, −5.08733708740123959921502195683, −3.78804869094843768702937987003, −3.64651413614662982297805207919, −2.63725641750577380128709317182, −1.07667475643668076962316281761, 0,
1.07667475643668076962316281761, 2.63725641750577380128709317182, 3.64651413614662982297805207919, 3.78804869094843768702937987003, 5.08733708740123959921502195683, 5.76565435697176124308807435048, 6.30707578029302709754962316307, 7.36633506980232662161209317627, 7.79557568480264821856777293184
