| L(s) = 1 | + 3-s − 5-s + 4·7-s + 9-s − 15-s − 17-s + 4·19-s + 4·21-s + 25-s + 27-s + 3·29-s + 3·31-s − 4·35-s + 10·37-s + 10·41-s − 10·43-s − 45-s + 6·47-s + 9·49-s − 51-s − 11·53-s + 4·57-s + 11·59-s + 4·61-s + 4·63-s + 5·67-s − 8·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.258·15-s − 0.242·17-s + 0.917·19-s + 0.872·21-s + 1/5·25-s + 0.192·27-s + 0.557·29-s + 0.538·31-s − 0.676·35-s + 1.64·37-s + 1.56·41-s − 1.52·43-s − 0.149·45-s + 0.875·47-s + 9/7·49-s − 0.140·51-s − 1.51·53-s + 0.529·57-s + 1.43·59-s + 0.512·61-s + 0.503·63-s + 0.610·67-s − 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.376570462\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.376570462\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.54998502767504, −12.02795025217240, −11.50352553807874, −11.37596293825492, −10.84734211172015, −10.26577110676136, −9.865167090337322, −9.225096406786501, −8.925382224159255, −8.254841819067969, −8.013328443970971, −7.650773765637755, −7.220385645828808, −6.552220382098049, −6.087255229062244, −5.349169184431400, −4.943810176377774, −4.509692444808516, −4.043434105066786, −3.504066301611498, −2.777341583550225, −2.415386865755123, −1.726570103710686, −1.109458372953634, −0.6464747406209582,
0.6464747406209582, 1.109458372953634, 1.726570103710686, 2.415386865755123, 2.777341583550225, 3.504066301611498, 4.043434105066786, 4.509692444808516, 4.943810176377774, 5.349169184431400, 6.087255229062244, 6.552220382098049, 7.220385645828808, 7.650773765637755, 8.013328443970971, 8.254841819067969, 8.925382224159255, 9.225096406786501, 9.865167090337322, 10.26577110676136, 10.84734211172015, 11.37596293825492, 11.50352553807874, 12.02795025217240, 12.54998502767504
