| L(s) = 1 | + 1.88·3-s + 2.42i·5-s − i·7-s + 0.548·9-s − i·11-s + (−3.47 + 0.973i)13-s + 4.56i·15-s − 6.01·17-s − 4.07i·19-s − 1.88i·21-s + 2.20·23-s − 0.861·25-s − 4.61·27-s − 3.84·29-s + 3.81i·31-s + ⋯ |
| L(s) = 1 | + 1.08·3-s + 1.08i·5-s − 0.377i·7-s + 0.182·9-s − 0.301i·11-s + (−0.962 + 0.269i)13-s + 1.17i·15-s − 1.45·17-s − 0.935i·19-s − 0.411i·21-s + 0.459·23-s − 0.172·25-s − 0.888·27-s − 0.714·29-s + 0.684i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.269i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 + 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1428442285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1428442285\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + iT \) |
| 13 | \( 1 + (3.47 - 0.973i)T \) |
| good | 3 | \( 1 - 1.88T + 3T^{2} \) |
| 5 | \( 1 - 2.42iT - 5T^{2} \) |
| 17 | \( 1 + 6.01T + 17T^{2} \) |
| 19 | \( 1 + 4.07iT - 19T^{2} \) |
| 23 | \( 1 - 2.20T + 23T^{2} \) |
| 29 | \( 1 + 3.84T + 29T^{2} \) |
| 31 | \( 1 - 3.81iT - 31T^{2} \) |
| 37 | \( 1 - 1.98iT - 37T^{2} \) |
| 41 | \( 1 + 5.43iT - 41T^{2} \) |
| 43 | \( 1 + 2.35T + 43T^{2} \) |
| 47 | \( 1 - 3.78iT - 47T^{2} \) |
| 53 | \( 1 + 8.79T + 53T^{2} \) |
| 59 | \( 1 - 5.32iT - 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 + 2.87iT - 67T^{2} \) |
| 71 | \( 1 - 1.02iT - 71T^{2} \) |
| 73 | \( 1 - 6.84iT - 73T^{2} \) |
| 79 | \( 1 + 4.13T + 79T^{2} \) |
| 83 | \( 1 + 1.29iT - 83T^{2} \) |
| 89 | \( 1 + 1.40iT - 89T^{2} \) |
| 97 | \( 1 + 7.82iT - 97T^{2} \) |
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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
−8.930927713231406551224772868690, −8.128550395501642598015055450565, −7.20686525947665717789306307496, −6.99673842128378986410595302083, −6.09700569558740017119166468876, −4.93244979056107031629422808282, −4.17016443759770736471857231379, −3.12501358090491679124998620285, −2.74409452570184346156122668311, −1.85139604229265429647339094859,
0.03038736959866491115692586587, 1.65925440161575071006815447372, 2.37792578831099340260304771192, 3.27769493827912819424852988604, 4.28635355239281989968719258509, 4.90423773574810005682049149564, 5.71374426289614947038007599444, 6.65780589517908734182009091881, 7.66867614653126090429987635283, 8.084842394760123985683792653533
