| L(s) = 1 | − i·2-s − 4-s − 4·7-s + i·8-s + 3·9-s + 2i·11-s + 2·13-s + 4i·14-s + 16-s − 6i·17-s − 3i·18-s + 2i·19-s + 2·22-s − 2i·26-s + 4·28-s + (5 + 2i)29-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 1.51·7-s + 0.353i·8-s + 9-s + 0.603i·11-s + 0.554·13-s + 1.06i·14-s + 0.250·16-s − 1.45i·17-s − 0.707i·18-s + 0.458i·19-s + 0.426·22-s − 0.392i·26-s + 0.755·28-s + (0.928 + 0.371i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.371 + 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.209397047\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.209397047\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 29 | \( 1 + (-5 - 2i)T \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 7 | \( 1 + 4T + 7T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 31 | \( 1 + 10iT - 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 12iT - 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 4iT - 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 10iT - 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + 12iT - 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−9.522242476177773885222778169512, −8.796684371423942440967441279508, −7.54516099766407673129006049633, −6.88810573446898751564299525664, −6.01987737866691909460284054105, −4.87813054722592132991634827308, −3.95296254505788711220411590717, −3.15899281988744824562927099714, −2.06129268661854677775419839084, −0.56237463239819935227973816108,
1.18630674931797418028301708653, 3.01520635953195875273714582466, 3.82405499877116359613216282805, 4.74879272076075981508326432880, 6.11840766913588807776105168338, 6.33091167843160679979236526602, 7.19737052362686052325325916873, 8.175731941488967674597448114890, 8.909551996272712047639285244018, 9.703441520681004803411733491392
