L(s) = 1 | − 0.930i·2-s − 1.32i·3-s + 1.13·4-s + (−2.23 − 0.0386i)5-s − 1.23·6-s − 4.33i·7-s − 2.91i·8-s + 1.24·9-s + (−0.0359 + 2.07i)10-s + 0.882·11-s − 1.50i·12-s − 2.68i·13-s − 4.03·14-s + (−0.0512 + 2.96i)15-s − 0.442·16-s + 0.304i·17-s + ⋯ |
L(s) = 1 | − 0.657i·2-s − 0.765i·3-s + 0.567·4-s + (−0.999 − 0.0172i)5-s − 0.503·6-s − 1.63i·7-s − 1.03i·8-s + 0.413·9-s + (−0.0113 + 0.657i)10-s + 0.266·11-s − 0.434i·12-s − 0.743i·13-s − 1.07·14-s + (−0.0132 + 0.765i)15-s − 0.110·16-s + 0.0738i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0172i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0172i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.654978065\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.654978065\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (2.23 + 0.0386i)T \) |
| 241 | \( 1 - T \) |
good | 2 | \( 1 + 0.930iT - 2T^{2} \) |
| 3 | \( 1 + 1.32iT - 3T^{2} \) |
| 7 | \( 1 + 4.33iT - 7T^{2} \) |
| 11 | \( 1 - 0.882T + 11T^{2} \) |
| 13 | \( 1 + 2.68iT - 13T^{2} \) |
| 17 | \( 1 - 0.304iT - 17T^{2} \) |
| 19 | \( 1 - 4.13T + 19T^{2} \) |
| 23 | \( 1 - 2.19iT - 23T^{2} \) |
| 29 | \( 1 + 8.97T + 29T^{2} \) |
| 31 | \( 1 - 5.91T + 31T^{2} \) |
| 37 | \( 1 - 10.9iT - 37T^{2} \) |
| 41 | \( 1 + 6.07T + 41T^{2} \) |
| 43 | \( 1 - 2.71iT - 43T^{2} \) |
| 47 | \( 1 - 9.50iT - 47T^{2} \) |
| 53 | \( 1 + 11.0iT - 53T^{2} \) |
| 59 | \( 1 - 7.93T + 59T^{2} \) |
| 61 | \( 1 + 1.60T + 61T^{2} \) |
| 67 | \( 1 + 15.5iT - 67T^{2} \) |
| 71 | \( 1 + 9.40T + 71T^{2} \) |
| 73 | \( 1 - 4.11iT - 73T^{2} \) |
| 79 | \( 1 + 9.90T + 79T^{2} \) |
| 83 | \( 1 - 10.7iT - 83T^{2} \) |
| 89 | \( 1 - 9.48T + 89T^{2} \) |
| 97 | \( 1 + 1.41iT - 97T^{2} \) |
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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
−9.737329092926165182576974020952, −8.127554096448996027505218717450, −7.55729223597961627617243998468, −7.08621196834159246441431470625, −6.34251555863254811282802505657, −4.74964373355275329373230311027, −3.76318724814692510564327322182, −3.15512618839718198885785375061, −1.51180179760057382904742454944, −0.74599834087566876078893960918,
1.98419345276183775287096380090, 3.14218528347521505853626561741, 4.20004819688069287535381391332, 5.20109036090849218275847261219, 5.87961095157721219724634087783, 6.99394416947257734895585878012, 7.53126495488492367313269806355, 8.697752896074209070078655977286, 9.008801239053248769227701585347, 10.08836383321975666255324249679