L(s) = 1 | + 3-s + 5-s + 4·7-s + 9-s − 2·13-s + 15-s − 6·17-s + 4·21-s + 4·23-s + 25-s + 27-s − 2·29-s + 8·31-s + 4·35-s + 6·37-s − 2·39-s − 6·41-s − 12·43-s + 45-s + 12·47-s + 9·49-s − 6·51-s − 10·53-s − 8·59-s − 10·61-s + 4·63-s − 2·65-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.554·13-s + 0.258·15-s − 1.45·17-s + 0.872·21-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.676·35-s + 0.986·37-s − 0.320·39-s − 0.937·41-s − 1.82·43-s + 0.149·45-s + 1.75·47-s + 9/7·49-s − 0.840·51-s − 1.37·53-s − 1.04·59-s − 1.28·61-s + 0.503·63-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.039594902\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.039594902\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−11.02833330477123621062994045589, −10.08392402285884229067283260571, −9.057592697379553614937900584363, −8.367348398622177159377234727845, −7.48722458989503568024470643880, −6.45809002818751957123652112981, −5.04896876370889379400771085477, −4.40699627501457718364255705443, −2.72073611368523068880201230919, −1.63558305614668413925749537297,
1.63558305614668413925749537297, 2.72073611368523068880201230919, 4.40699627501457718364255705443, 5.04896876370889379400771085477, 6.45809002818751957123652112981, 7.48722458989503568024470643880, 8.367348398622177159377234727845, 9.057592697379553614937900584363, 10.08392402285884229067283260571, 11.02833330477123621062994045589