L(s) = 1 | + (−3.30 − 5.73i)5-s + (4.31 − 18.0i)7-s + (−1.54 − 0.893i)11-s + 4.72i·13-s + (23.8 − 41.3i)17-s + (40.1 − 23.1i)19-s + (−30.2 + 17.4i)23-s + (40.5 − 70.3i)25-s − 48.3i·29-s + (107. + 62.2i)31-s + (−117. + 34.8i)35-s + (−137. − 238. i)37-s + 37.3·41-s + 215.·43-s + (53.1 + 91.9i)47-s + ⋯ |
L(s) = 1 | + (−0.295 − 0.512i)5-s + (0.233 − 0.972i)7-s + (−0.0424 − 0.0244i)11-s + 0.100i·13-s + (0.340 − 0.589i)17-s + (0.484 − 0.279i)19-s + (−0.273 + 0.158i)23-s + (0.324 − 0.562i)25-s − 0.309i·29-s + (0.624 + 0.360i)31-s + (−0.567 + 0.168i)35-s + (−0.612 − 1.06i)37-s + 0.142·41-s + 0.765·43-s + (0.164 + 0.285i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.874 + 0.485i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.874 + 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.286786281\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.286786281\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-4.31 + 18.0i)T \) |
good | 5 | \( 1 + (3.30 + 5.73i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (1.54 + 0.893i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 4.72iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-23.8 + 41.3i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-40.1 + 23.1i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (30.2 - 17.4i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 48.3iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-107. - 62.2i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (137. + 238. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 37.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 215.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-53.1 - 91.9i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (233. + 134. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (149. - 259. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (292. - 168. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-188. + 326. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 816. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (596. + 344. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-307. - 531. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 1.32e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (480. + 832. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 449. iT - 9.12e5T^{2} \) |
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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
−9.246628921886140091667686452365, −8.338743420367952688499156889720, −7.56268318965291582065216554435, −6.87796559801147406441257186906, −5.70025532604837996698781586250, −4.71450728747696838361008757139, −4.01670652418798152751128332014, −2.83782995889494577532676049472, −1.33211228109544939000276061075, −0.33914193591318873738982125261,
1.42985378586544638513203112773, 2.66611329640925517641385066647, 3.53671568776723822966086165836, 4.76763774714029611854698317889, 5.69085702736690483143425006785, 6.47890070042712917997871365137, 7.52504309563918538347826576429, 8.238709610016819796656140785936, 9.062753889794020339882687683254, 9.944397412995400272961593114345