| L(s) = 1 | − i·2-s − 4-s + (1.41 + 1.73i)5-s − 3.86i·7-s + i·8-s + (1.73 − 1.41i)10-s − 1.03·11-s − 3.86i·13-s − 3.86·14-s + 16-s + 0.535i·17-s + 19-s + (−1.41 − 1.73i)20-s + 1.03i·22-s − 5.46i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (0.632 + 0.774i)5-s − 1.46i·7-s + 0.353i·8-s + (0.547 − 0.447i)10-s − 0.312·11-s − 1.07i·13-s − 1.03·14-s + 0.250·16-s + 0.129i·17-s + 0.229·19-s + (−0.316 − 0.387i)20-s + 0.220i·22-s − 1.13i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.774 + 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.384933833\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.384933833\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.41 - 1.73i)T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + 3.86iT - 7T^{2} \) |
| 11 | \( 1 + 1.03T + 11T^{2} \) |
| 13 | \( 1 + 3.86iT - 13T^{2} \) |
| 17 | \( 1 - 0.535iT - 17T^{2} \) |
| 23 | \( 1 + 5.46iT - 23T^{2} \) |
| 29 | \( 1 - 4.62T + 29T^{2} \) |
| 31 | \( 1 + 10.9T + 31T^{2} \) |
| 37 | \( 1 - 1.79iT - 37T^{2} \) |
| 41 | \( 1 + 1.79T + 41T^{2} \) |
| 43 | \( 1 + 6.69iT - 43T^{2} \) |
| 47 | \( 1 - 2.53iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 6.96T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 13.3iT - 67T^{2} \) |
| 71 | \( 1 + 4.14T + 71T^{2} \) |
| 73 | \( 1 + 3.58iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 16.3iT - 83T^{2} \) |
| 89 | \( 1 + 3.86T + 89T^{2} \) |
| 97 | \( 1 + 2.55iT - 97T^{2} \) |
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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
−9.270159720496802452699148473961, −8.218269413612367199183606943727, −7.42579639595462030372841927360, −6.71882629211295435896227598438, −5.70046502246636821945486655174, −4.78708559987043148784141982352, −3.70487931564221454200154951013, −3.02420559436866091891460811193, −1.87323763132524024586236185249, −0.52089821926849742015954871748,
1.54740250858841140582978623757, 2.58878849398296448947193872220, 3.99716309926727561057670723579, 5.11350153486618061737082230711, 5.51899436841597340722107666842, 6.28186621316383995923396843045, 7.22594556032278901936825673558, 8.198332462743925799730183673153, 8.970014556695049075911467089032, 9.273662417570243393978233755057
