| L(s) = 1 | + i·5-s + 0.146·7-s + 2.68i·11-s + i·13-s − 17-s − 4i·19-s − 6.68·23-s + 4·25-s + 4.39i·29-s − 1.31·31-s + 0.146i·35-s + 3.97i·37-s + 6.39·41-s + 6.83i·43-s − 7.12·47-s + ⋯ |
| L(s) = 1 | + 0.447i·5-s + 0.0553·7-s + 0.809i·11-s + 0.277i·13-s − 0.242·17-s − 0.917i·19-s − 1.39·23-s + 0.800·25-s + 0.815i·29-s − 0.236·31-s + 0.0247i·35-s + 0.654i·37-s + 0.998·41-s + 1.04i·43-s − 1.03·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 - 0.474i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.880 - 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8242197465\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8242197465\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - iT \) |
| good | 5 | \( 1 - iT - 5T^{2} \) |
| 7 | \( 1 - 0.146T + 7T^{2} \) |
| 11 | \( 1 - 2.68iT - 11T^{2} \) |
| 17 | \( 1 + T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 6.68T + 23T^{2} \) |
| 29 | \( 1 - 4.39iT - 29T^{2} \) |
| 31 | \( 1 + 1.31T + 31T^{2} \) |
| 37 | \( 1 - 3.97iT - 37T^{2} \) |
| 41 | \( 1 - 6.39T + 41T^{2} \) |
| 43 | \( 1 - 6.83iT - 43T^{2} \) |
| 47 | \( 1 + 7.12T + 47T^{2} \) |
| 53 | \( 1 - 8.97iT - 53T^{2} \) |
| 59 | \( 1 + 12.3iT - 59T^{2} \) |
| 61 | \( 1 - 8.35iT - 61T^{2} \) |
| 67 | \( 1 + 8.29iT - 67T^{2} \) |
| 71 | \( 1 - 5.51T + 71T^{2} \) |
| 73 | \( 1 + 6.97T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 - 4.29iT - 83T^{2} \) |
| 89 | \( 1 - 5.37T + 89T^{2} \) |
| 97 | \( 1 + 10.3T + 97T^{2} \) |
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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
−8.828047687063215560238541461735, −8.044345249793144742701000666309, −7.27213144617943707343036344744, −6.67763328739481918733274049510, −5.98530788204705242714317555993, −4.88009162144863544674267465057, −4.38722828708121773019071104337, −3.28468885753482482673067769556, −2.45375308120817373834943553929, −1.44971409191930854418619805479,
0.23620846614811677351106738692, 1.48892661283108289942182518434, 2.55871829757356654432196465352, 3.62745367643352499401562917504, 4.27832328279734165301127903110, 5.32070897237411174092760298261, 5.88772294076835178532504478581, 6.63255007738218567187327687126, 7.65736277625808939025577319397, 8.230214611412597880447749712724
