L(s) = 1 | + (−0.989 − 0.142i)2-s + (0.909 − 0.415i)3-s + (0.959 + 0.281i)4-s + (−0.959 + 0.281i)6-s + (0.540 − 0.841i)7-s + (−0.909 − 0.415i)8-s + (0.654 − 0.755i)9-s + (0.142 + 0.989i)11-s + (0.989 − 0.142i)12-s + (−0.540 − 0.841i)13-s + (−0.654 + 0.755i)14-s + (0.841 + 0.540i)16-s + (0.281 + 0.959i)17-s + (−0.755 + 0.654i)18-s + (−0.959 − 0.281i)19-s + ⋯ |
L(s) = 1 | + (−0.989 − 0.142i)2-s + (0.909 − 0.415i)3-s + (0.959 + 0.281i)4-s + (−0.959 + 0.281i)6-s + (0.540 − 0.841i)7-s + (−0.909 − 0.415i)8-s + (0.654 − 0.755i)9-s + (0.142 + 0.989i)11-s + (0.989 − 0.142i)12-s + (−0.540 − 0.841i)13-s + (−0.654 + 0.755i)14-s + (0.841 + 0.540i)16-s + (0.281 + 0.959i)17-s + (−0.755 + 0.654i)18-s + (−0.959 − 0.281i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.660 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.660 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8694626792 - 0.3932016493i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8694626792 - 0.3932016493i\) |
\(L(1)\) |
\(\approx\) |
\(0.9164537822 - 0.2515906542i\) |
\(L(1)\) |
\(\approx\) |
\(0.9164537822 - 0.2515906542i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + (-0.989 - 0.142i)T \) |
| 3 | \( 1 + (0.909 - 0.415i)T \) |
| 7 | \( 1 + (0.540 - 0.841i)T \) |
| 11 | \( 1 + (0.142 + 0.989i)T \) |
| 13 | \( 1 + (-0.540 - 0.841i)T \) |
| 17 | \( 1 + (0.281 + 0.959i)T \) |
| 19 | \( 1 + (-0.959 - 0.281i)T \) |
| 29 | \( 1 + (0.959 - 0.281i)T \) |
| 31 | \( 1 + (0.415 - 0.909i)T \) |
| 37 | \( 1 + (0.755 + 0.654i)T \) |
| 41 | \( 1 + (-0.654 - 0.755i)T \) |
| 43 | \( 1 + (-0.909 + 0.415i)T \) |
| 47 | \( 1 + iT \) |
| 53 | \( 1 + (-0.540 + 0.841i)T \) |
| 59 | \( 1 + (-0.841 + 0.540i)T \) |
| 61 | \( 1 + (-0.415 + 0.909i)T \) |
| 67 | \( 1 + (0.989 + 0.142i)T \) |
| 71 | \( 1 + (-0.142 + 0.989i)T \) |
| 73 | \( 1 + (-0.281 + 0.959i)T \) |
| 79 | \( 1 + (0.841 - 0.540i)T \) |
| 83 | \( 1 + (-0.755 - 0.654i)T \) |
| 89 | \( 1 + (0.415 + 0.909i)T \) |
| 97 | \( 1 + (-0.755 + 0.654i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−29.27576167450832828113385397773, −28.13060915270144626283741619835, −27.10541221524557860854560944668, −26.68013877666362745575891447586, −25.28032162277294495395627601022, −24.866516448760993029923721143622, −23.74078842522074811726279281007, −21.63759958927262408308853019604, −21.23194685235200732402978679179, −19.92570049612389676999965630108, −19.03717081106329787092545877519, −18.29599559745132649562213511374, −16.78461656108867334112036124322, −15.92000011782317600845700503438, −14.84145181038762540991908393249, −14.00916712229031535426135148898, −12.12650894604414077973051030136, −11.0070996076235675803018574303, −9.72640584563349213455630906856, −8.780217125157839056552536672787, −8.071614427245782656743130766750, −6.62050869083954223890536049354, −4.98651830979259253733518907458, −3.05553626557108661557684707681, −1.87659792705625067946234597722,
1.35935495493342975762268341344, 2.63148708374185400467866722854, 4.212490045634931749103453321454, 6.54864059029530069314503404892, 7.64438336852594964825485760586, 8.31723914683589480375771447760, 9.735538400734983142373779787385, 10.54564311775213465142881928352, 12.119111673466837265675171035513, 13.12907104904845266012932469480, 14.68424711236582380699786912134, 15.34689728179198392477184050941, 17.10564717216873379902281835683, 17.66400663006251668435018277372, 18.92402067827591210706520083563, 19.8960228712334716644856702869, 20.449243932780904198109635448245, 21.51793202246640004964653551436, 23.36821764503148520751854098729, 24.36916357136370930028006292590, 25.36730515862420039222636273359, 26.07557398465779278391825074742, 27.09057228346832598446913067475, 27.88148217654275451384492021039, 29.24868316281304453222654483777