| L(s) = 1 | − 5·7-s − 11-s + 13-s + 17-s − 4·19-s − 6·23-s − 3·29-s − 5·31-s − 6·37-s + 8·41-s − 2·43-s + 3·47-s + 18·49-s − 5·53-s + 9·59-s − 15·61-s − 7·67-s + 8·71-s + 2·73-s + 5·77-s − 10·79-s + 3·83-s − 5·91-s + 10·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.88·7-s − 0.301·11-s + 0.277·13-s + 0.242·17-s − 0.917·19-s − 1.25·23-s − 0.557·29-s − 0.898·31-s − 0.986·37-s + 1.24·41-s − 0.304·43-s + 0.437·47-s + 18/7·49-s − 0.686·53-s + 1.17·59-s − 1.92·61-s − 0.855·67-s + 0.949·71-s + 0.234·73-s + 0.569·77-s − 1.12·79-s + 0.329·83-s − 0.524·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−14.22521377748372, −13.72768671259966, −13.29898885800440, −12.76932807317340, −12.47724663618158, −12.08044163193516, −11.30966394703893, −10.75371088414261, −10.31291257225647, −9.894575134082147, −9.356422658843715, −8.938396512369904, −8.423235446695508, −7.606365641523320, −7.338705566318109, −6.532269452709686, −6.236034335224590, −5.773606095661452, −5.183257709031596, −4.278023335538958, −3.786279898013154, −3.390913563245619, −2.643510343131143, −2.138458391931488, −1.229786912982615, 0, 0,
1.229786912982615, 2.138458391931488, 2.643510343131143, 3.390913563245619, 3.786279898013154, 4.278023335538958, 5.183257709031596, 5.773606095661452, 6.236034335224590, 6.532269452709686, 7.338705566318109, 7.606365641523320, 8.423235446695508, 8.938396512369904, 9.356422658843715, 9.894575134082147, 10.31291257225647, 10.75371088414261, 11.30966394703893, 12.08044163193516, 12.47724663618158, 12.76932807317340, 13.29898885800440, 13.72768671259966, 14.22521377748372
