| L(s) = 1 | − 2.37·5-s + 7-s − 1.37·13-s + 17-s + 6.74·19-s − 5.37·23-s + 0.627·25-s + 0.627·29-s + 2.62·31-s − 2.37·35-s + 4.37·37-s + 5.11·41-s + 7.74·43-s + 0.372·47-s + 49-s + 12.1·53-s − 0.255·59-s + 1.25·61-s + 3.25·65-s + 4.62·67-s − 7.37·71-s + 10·73-s − 9.11·79-s + 15.8·83-s − 2.37·85-s + 2.62·89-s − 1.37·91-s + ⋯ |
| L(s) = 1 | − 1.06·5-s + 0.377·7-s − 0.380·13-s + 0.242·17-s + 1.54·19-s − 1.12·23-s + 0.125·25-s + 0.116·29-s + 0.471·31-s − 0.400·35-s + 0.718·37-s + 0.799·41-s + 1.18·43-s + 0.0543·47-s + 0.142·49-s + 1.66·53-s − 0.0332·59-s + 0.160·61-s + 0.403·65-s + 0.565·67-s − 0.874·71-s + 1.17·73-s − 1.02·79-s + 1.74·83-s − 0.257·85-s + 0.278·89-s − 0.143·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.377405778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.377405778\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + 2.37T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 1.37T + 13T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 19 | \( 1 - 6.74T + 19T^{2} \) |
| 23 | \( 1 + 5.37T + 23T^{2} \) |
| 29 | \( 1 - 0.627T + 29T^{2} \) |
| 31 | \( 1 - 2.62T + 31T^{2} \) |
| 37 | \( 1 - 4.37T + 37T^{2} \) |
| 41 | \( 1 - 5.11T + 41T^{2} \) |
| 43 | \( 1 - 7.74T + 43T^{2} \) |
| 47 | \( 1 - 0.372T + 47T^{2} \) |
| 53 | \( 1 - 12.1T + 53T^{2} \) |
| 59 | \( 1 + 0.255T + 59T^{2} \) |
| 61 | \( 1 - 1.25T + 61T^{2} \) |
| 67 | \( 1 - 4.62T + 67T^{2} \) |
| 71 | \( 1 + 7.37T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 9.11T + 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 - 2.62T + 89T^{2} \) |
| 97 | \( 1 - 9.48T + 97T^{2} \) |
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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
−9.494371253326006661345522740413, −8.558753907729200267060310096879, −7.61484721121862250599219428100, −7.50519700112715179141567584728, −6.18241795578000492124680060514, −5.27426583044841362411508059967, −4.32669599270316909629217427887, −3.57178027611384620479456878112, −2.41648034852488218705625360947, −0.842275583713282463176159217319,
0.842275583713282463176159217319, 2.41648034852488218705625360947, 3.57178027611384620479456878112, 4.32669599270316909629217427887, 5.27426583044841362411508059967, 6.18241795578000492124680060514, 7.50519700112715179141567584728, 7.61484721121862250599219428100, 8.558753907729200267060310096879, 9.494371253326006661345522740413
