| L(s) = 1 | − 5-s − 2·13-s + 6·17-s − 4·19-s + 25-s + 6·29-s + 8·31-s + 2·37-s − 6·41-s + 4·43-s + 6·53-s + 10·61-s + 2·65-s + 4·67-s − 2·73-s − 8·79-s − 12·83-s − 6·85-s + 18·89-s + 4·95-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.554·13-s + 1.45·17-s − 0.917·19-s + 1/5·25-s + 1.11·29-s + 1.43·31-s + 0.328·37-s − 0.937·41-s + 0.609·43-s + 0.824·53-s + 1.28·61-s + 0.248·65-s + 0.488·67-s − 0.234·73-s − 0.900·79-s − 1.31·83-s − 0.650·85-s + 1.90·89-s + 0.410·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.022255060\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.022255060\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.83273536247524, −14.45204857149954, −14.04824925444087, −13.28837000396352, −12.82910376082936, −12.16167835501519, −11.90984143334038, −11.37770116332942, −10.55361320298529, −10.14098473368285, −9.807048303172031, −8.920673595966519, −8.422528671641072, −7.945550945652287, −7.394956172154084, −6.726714288580323, −6.230798408735269, −5.474526893497198, −4.897689698831675, −4.283484900065737, −3.659551036903420, −2.892171404917692, −2.364840434899152, −1.302443862150284, −0.5724796509798254,
0.5724796509798254, 1.302443862150284, 2.364840434899152, 2.892171404917692, 3.659551036903420, 4.283484900065737, 4.897689698831675, 5.474526893497198, 6.230798408735269, 6.726714288580323, 7.394956172154084, 7.945550945652287, 8.422528671641072, 8.920673595966519, 9.807048303172031, 10.14098473368285, 10.55361320298529, 11.37770116332942, 11.90984143334038, 12.16167835501519, 12.82910376082936, 13.28837000396352, 14.04824925444087, 14.45204857149954, 14.83273536247524
