| L(s) = 1 | − 0.198i·3-s − 1.31i·5-s + (−0.640 − 2.56i)7-s + 2.96·9-s + (1.56 + 2.92i)11-s − 13-s − 0.260·15-s − 4.67·17-s − 4.13·19-s + (−0.509 + 0.127i)21-s + 0.678·23-s + 3.28·25-s − 1.18i·27-s − 4.73i·29-s − 3.43i·31-s + ⋯ |
| L(s) = 1 | − 0.114i·3-s − 0.586i·5-s + (−0.241 − 0.970i)7-s + 0.986·9-s + (0.471 + 0.881i)11-s − 0.277·13-s − 0.0672·15-s − 1.13·17-s − 0.948·19-s + (−0.111 + 0.0277i)21-s + 0.141·23-s + 0.656·25-s − 0.227i·27-s − 0.879i·29-s − 0.616i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.741 + 0.671i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.741 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.281727198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.281727198\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (0.640 + 2.56i)T \) |
| 11 | \( 1 + (-1.56 - 2.92i)T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 0.198iT - 3T^{2} \) |
| 5 | \( 1 + 1.31iT - 5T^{2} \) |
| 17 | \( 1 + 4.67T + 17T^{2} \) |
| 19 | \( 1 + 4.13T + 19T^{2} \) |
| 23 | \( 1 - 0.678T + 23T^{2} \) |
| 29 | \( 1 + 4.73iT - 29T^{2} \) |
| 31 | \( 1 + 3.43iT - 31T^{2} \) |
| 37 | \( 1 + 1.92T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 + 10.0iT - 43T^{2} \) |
| 47 | \( 1 + 2.08iT - 47T^{2} \) |
| 53 | \( 1 + 8.71T + 53T^{2} \) |
| 59 | \( 1 + 7.26iT - 59T^{2} \) |
| 61 | \( 1 + 1.49T + 61T^{2} \) |
| 67 | \( 1 + 4.45T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 + 9.55T + 73T^{2} \) |
| 79 | \( 1 + 1.30iT - 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 + 9.83iT - 89T^{2} \) |
| 97 | \( 1 - 14.3iT - 97T^{2} \) |
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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
−8.085249656943083548001035758621, −7.31089629429843759410887497616, −6.83335791816467487551579206749, −6.15326526533817018212451729538, −4.85151933198199066249000351570, −4.36209935166376662875075275749, −3.85785527919512709615179302951, −2.38550166884685763872388328282, −1.52893363189749298633773970435, −0.36887345878647942341764029992,
1.36941210452419276885222448428, 2.49470683377637354474008341179, 3.18033913141444123132569217917, 4.21544907619299749347751680331, 4.89471092172077291920436835402, 5.96815986934293920045474035757, 6.52272504661107726605345657973, 7.08338124719946396733199161210, 8.066302397651282476180514849993, 8.919995242700279653284357899824
