L(s) = 1 | + (−0.920 + 0.389i)3-s + (0.630 − 0.776i)5-s + (0.517 − 0.855i)7-s + (0.695 − 0.718i)9-s + (−0.930 − 0.366i)11-s + (0.971 − 0.235i)13-s + (−0.277 + 0.960i)15-s + (0.764 + 0.644i)17-s + (−0.992 + 0.124i)19-s + (−0.143 + 0.989i)21-s + (−0.943 + 0.331i)23-s + (−0.205 − 0.978i)25-s + (−0.360 + 0.932i)27-s + (0.441 + 0.897i)29-s + (0.747 + 0.663i)31-s + ⋯ |
L(s) = 1 | + (−0.920 + 0.389i)3-s + (0.630 − 0.776i)5-s + (0.517 − 0.855i)7-s + (0.695 − 0.718i)9-s + (−0.930 − 0.366i)11-s + (0.971 − 0.235i)13-s + (−0.277 + 0.960i)15-s + (0.764 + 0.644i)17-s + (−0.992 + 0.124i)19-s + (−0.143 + 0.989i)21-s + (−0.943 + 0.331i)23-s + (−0.205 − 0.978i)25-s + (−0.360 + 0.932i)27-s + (0.441 + 0.897i)29-s + (0.747 + 0.663i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.376 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.376 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2872219670 + 0.4265911272i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2872219670 + 0.4265911272i\) |
\(L(1)\) |
\(\approx\) |
\(0.7995133502 - 0.03242321984i\) |
\(L(1)\) |
\(\approx\) |
\(0.7995133502 - 0.03242321984i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 503 | \( 1 \) |
good | 3 | \( 1 + (-0.920 + 0.389i)T \) |
| 5 | \( 1 + (0.630 - 0.776i)T \) |
| 7 | \( 1 + (0.517 - 0.855i)T \) |
| 11 | \( 1 + (-0.930 - 0.366i)T \) |
| 13 | \( 1 + (0.971 - 0.235i)T \) |
| 17 | \( 1 + (0.764 + 0.644i)T \) |
| 19 | \( 1 + (-0.992 + 0.124i)T \) |
| 23 | \( 1 + (-0.943 + 0.331i)T \) |
| 29 | \( 1 + (0.441 + 0.897i)T \) |
| 31 | \( 1 + (0.747 + 0.663i)T \) |
| 37 | \( 1 + (-0.984 - 0.174i)T \) |
| 41 | \( 1 + (-0.0812 + 0.996i)T \) |
| 43 | \( 1 + (0.507 + 0.861i)T \) |
| 47 | \( 1 + (0.265 + 0.964i)T \) |
| 53 | \( 1 + (-0.992 - 0.124i)T \) |
| 59 | \( 1 + (-0.452 + 0.891i)T \) |
| 61 | \( 1 + (-0.999 - 0.0250i)T \) |
| 67 | \( 1 + (0.277 + 0.960i)T \) |
| 71 | \( 1 + (-0.982 - 0.186i)T \) |
| 73 | \( 1 + (-0.958 + 0.283i)T \) |
| 79 | \( 1 + (-0.372 + 0.928i)T \) |
| 83 | \( 1 + (-0.998 + 0.0500i)T \) |
| 89 | \( 1 + (-0.00625 - 0.999i)T \) |
| 97 | \( 1 + (0.939 - 0.343i)T \) |
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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
−18.31273308790333484390084660542, −17.571859156137635806462110533575, −17.24296418234222715469274444589, −16.15123094204065414506763410698, −15.58417736150465963127771840482, −15.00840692745727785590717067150, −13.87676647996045010261302166028, −13.66799982471266455308563667708, −12.636187294761205379872918470957, −12.05369213595185215406817111282, −11.420815781963255489326504435396, −10.59946434977026741551917644356, −10.283975918516829406659380721309, −9.36030566312783960852723037090, −8.34942984298792798090286029644, −7.736862870614455085247493708072, −6.88184179205086349766141628896, −6.125168846126059562746554448389, −5.72059430468838489622354599467, −4.97729868748695238900673983294, −4.12393702513234689351213408184, −2.87972558728215555413245654365, −2.14510387536367492268784093630, −1.60432969900706658209361095391, −0.16111563117266406928308354487,
1.26824882801341609279465616686, 1.39447587115234129396989116022, 2.93251317655003975853752716045, 3.93370178028115771185925064994, 4.54902862424734898517927484032, 5.24165489953655953952254080330, 5.966117704485892012226847251583, 6.41272718781753601526670332646, 7.60145895992588659869276224124, 8.26644590052572895166535557045, 8.92742132819983000412782182068, 10.08982227959737055070057027089, 10.35646425359361901866060842775, 10.93643044127425919378807997871, 11.78190781127277623135625444697, 12.68043330786360411053696956368, 13.01044201099428115958102518270, 13.90390876131638318297502109560, 14.487334474666454196430406921924, 15.620630666355440626125072112663, 16.07779047623664517193591188246, 16.66752814134130339567335665240, 17.34859454285768633469332924045, 17.79935688713303118457198885640, 18.399729891693053204898896643907