| L(s) = 1 | + 0.481i·3-s + (1.67 + 1.48i)5-s + 0.806i·7-s + 2.76·9-s + 3.67·11-s − i·13-s + (−0.712 + 0.806i)15-s + 1.35i·17-s − 1.67·19-s − 0.387·21-s − 6.48i·23-s + (0.612 + 4.96i)25-s + 2.77i·27-s − 2.41·29-s + 5.28·31-s + ⋯ |
| L(s) = 1 | + 0.277i·3-s + (0.749 + 0.662i)5-s + 0.304i·7-s + 0.922·9-s + 1.10·11-s − 0.277i·13-s + (−0.184 + 0.208i)15-s + 0.327i·17-s − 0.384·19-s − 0.0846·21-s − 1.35i·23-s + (0.122 + 0.992i)25-s + 0.534i·27-s − 0.449·29-s + 0.949·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.079475258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.079475258\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.67 - 1.48i)T \) |
| 13 | \( 1 + iT \) |
| good | 3 | \( 1 - 0.481iT - 3T^{2} \) |
| 7 | \( 1 - 0.806iT - 7T^{2} \) |
| 11 | \( 1 - 3.67T + 11T^{2} \) |
| 17 | \( 1 - 1.35iT - 17T^{2} \) |
| 19 | \( 1 + 1.67T + 19T^{2} \) |
| 23 | \( 1 + 6.48iT - 23T^{2} \) |
| 29 | \( 1 + 2.41T + 29T^{2} \) |
| 31 | \( 1 - 5.28T + 31T^{2} \) |
| 37 | \( 1 + 3.76iT - 37T^{2} \) |
| 41 | \( 1 + 8.31T + 41T^{2} \) |
| 43 | \( 1 - 6.79iT - 43T^{2} \) |
| 47 | \( 1 - 3.19iT - 47T^{2} \) |
| 53 | \( 1 - 5.73iT - 53T^{2} \) |
| 59 | \( 1 - 5.98T + 59T^{2} \) |
| 61 | \( 1 + 1.76T + 61T^{2} \) |
| 67 | \( 1 - 9.89iT - 67T^{2} \) |
| 71 | \( 1 + 8.56T + 71T^{2} \) |
| 73 | \( 1 + 11.7iT - 73T^{2} \) |
| 79 | \( 1 + 2.26T + 79T^{2} \) |
| 83 | \( 1 + 3.84iT - 83T^{2} \) |
| 89 | \( 1 + 2.77T + 89T^{2} \) |
| 97 | \( 1 - 1.87iT - 97T^{2} \) |
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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
−10.13379277316794797665365267622, −9.297130199821374525774455545737, −8.579890947106659067012014864418, −7.36942566566821325215453980137, −6.56149523530080531488241383977, −5.96455636262384026404962775689, −4.73193693977314403066290682334, −3.84251925168766387061947782554, −2.64292719887928130446488022466, −1.47100908560065327134911604304,
1.13469122716665794176341484181, 1.99938546319313035491945104617, 3.67265261458423998500899151547, 4.54617287573423746095092440702, 5.51831730946595173743683503384, 6.56600789389474782002588093880, 7.11795810908392735664344655447, 8.241067785365236938699942732725, 9.098969085996931658112988970508, 9.764433887185709165679191095262
