| L(s) = 1 | − 3·3-s + 2·7-s + 6·9-s + 5·13-s + 6·17-s + 6·19-s − 6·21-s − 23-s − 9·27-s + 9·29-s + 3·31-s + 8·37-s − 15·39-s + 3·41-s + 8·43-s − 7·47-s − 3·49-s − 18·51-s + 2·53-s − 18·57-s + 4·59-s − 10·61-s + 12·63-s − 8·67-s + 3·69-s + 7·71-s − 9·73-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.755·7-s + 2·9-s + 1.38·13-s + 1.45·17-s + 1.37·19-s − 1.30·21-s − 0.208·23-s − 1.73·27-s + 1.67·29-s + 0.538·31-s + 1.31·37-s − 2.40·39-s + 0.468·41-s + 1.21·43-s − 1.02·47-s − 3/7·49-s − 2.52·51-s + 0.274·53-s − 2.38·57-s + 0.520·59-s − 1.28·61-s + 1.51·63-s − 0.977·67-s + 0.361·69-s + 0.830·71-s − 1.05·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.568026343\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.568026343\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−7.997402643331904740009462370615, −7.67899822443948661188579997050, −6.58275003853327185223548469271, −6.06483814765580990692180218296, −5.44663656641282997507750663047, −4.81358070969442574628445235427, −4.04667320938686712108909524918, −2.98213621017603004203348344774, −1.30055558518431425089443698022, −0.950133350562753842827187537167,
0.950133350562753842827187537167, 1.30055558518431425089443698022, 2.98213621017603004203348344774, 4.04667320938686712108909524918, 4.81358070969442574628445235427, 5.44663656641282997507750663047, 6.06483814765580990692180218296, 6.58275003853327185223548469271, 7.67899822443948661188579997050, 7.997402643331904740009462370615
