L(s) = 1 | + (−0.603 + 0.797i)2-s + (0.994 − 0.104i)3-s + (−0.272 − 0.962i)4-s + (−0.516 + 0.856i)6-s + (0.931 + 0.362i)8-s + (0.978 − 0.207i)9-s + (−0.371 − 0.928i)12-s + (−0.441 + 0.897i)13-s + (−0.851 + 0.524i)16-s + (−0.976 + 0.217i)17-s + (−0.424 + 0.905i)18-s + (0.749 − 0.662i)19-s + (0.971 − 0.235i)23-s + (0.964 + 0.263i)24-s + (−0.449 − 0.893i)26-s + (0.951 − 0.309i)27-s + ⋯ |
L(s) = 1 | + (−0.603 + 0.797i)2-s + (0.994 − 0.104i)3-s + (−0.272 − 0.962i)4-s + (−0.516 + 0.856i)6-s + (0.931 + 0.362i)8-s + (0.978 − 0.207i)9-s + (−0.371 − 0.928i)12-s + (−0.441 + 0.897i)13-s + (−0.851 + 0.524i)16-s + (−0.976 + 0.217i)17-s + (−0.424 + 0.905i)18-s + (0.749 − 0.662i)19-s + (0.971 − 0.235i)23-s + (0.964 + 0.263i)24-s + (−0.449 − 0.893i)26-s + (0.951 − 0.309i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.436094880 + 1.080078653i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.436094880 + 1.080078653i\) |
\(L(1)\) |
\(\approx\) |
\(1.051134745 + 0.3791240843i\) |
\(L(1)\) |
\(\approx\) |
\(1.051134745 + 0.3791240843i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.603 + 0.797i)T \) |
| 3 | \( 1 + (0.994 - 0.104i)T \) |
| 13 | \( 1 + (-0.441 + 0.897i)T \) |
| 17 | \( 1 + (-0.976 + 0.217i)T \) |
| 19 | \( 1 + (0.749 - 0.662i)T \) |
| 23 | \( 1 + (0.971 - 0.235i)T \) |
| 29 | \( 1 + (0.870 - 0.491i)T \) |
| 31 | \( 1 + (-0.640 + 0.768i)T \) |
| 37 | \( 1 + (0.0380 + 0.999i)T \) |
| 41 | \( 1 + (-0.974 - 0.226i)T \) |
| 43 | \( 1 + (-0.540 - 0.841i)T \) |
| 47 | \( 1 + (0.424 + 0.905i)T \) |
| 53 | \( 1 + (-0.524 + 0.851i)T \) |
| 59 | \( 1 + (-0.290 + 0.956i)T \) |
| 61 | \( 1 + (-0.123 - 0.992i)T \) |
| 67 | \( 1 + (0.189 + 0.981i)T \) |
| 71 | \( 1 + (0.993 + 0.113i)T \) |
| 73 | \( 1 + (-0.132 - 0.991i)T \) |
| 79 | \( 1 + (0.935 + 0.353i)T \) |
| 83 | \( 1 + (0.825 - 0.564i)T \) |
| 89 | \( 1 + (0.580 - 0.814i)T \) |
| 97 | \( 1 + (-0.967 + 0.254i)T \) |
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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
−18.16742318196844272938656898633, −17.96562444135908561925396202451, −16.94011798133877260819686177293, −16.27666005215303563658673832947, −15.50624484276131512863295840130, −14.87929711829135023299779389445, −14.006435960739367523732933107791, −13.367824960283646734386391910351, −12.79089477522855185428027706590, −12.16839255439466176362264558785, −11.206422365354074801288756225978, −10.61982907182668428165868454247, −9.80215357606694760457501689664, −9.41179675528232299382333820584, −8.58928622216948874229766844389, −8.0449838157387881487218214932, −7.35294458073441241663559286371, −6.70516437624330411545506857556, −5.27215601745768792109870916220, −4.60783153160237595516457668730, −3.61742631178186966558558813632, −3.15518584846903904538353741021, −2.35689792610754094383661849119, −1.65709133624125054922743205429, −0.633155574796526773918473237723,
0.91506646105011674060040756378, 1.798926417141267102567709773655, 2.54775698741910264476352497916, 3.52256732961359788727166893028, 4.655004984551150391831603242834, 4.895388008482172373159565139951, 6.25743331101775196102974053115, 6.86543639939515726106281576765, 7.331885893972597725514659953960, 8.175778621577100169987766772164, 8.94492514582158555649067143263, 9.17249643691416241043116732959, 10.0331587694393565714426037892, 10.71919938952339684742374426394, 11.60130305428849193566933974473, 12.53522609375354599564343732373, 13.51775868472645529363808428246, 13.79696315886602777010307296624, 14.548315052594803062289758012159, 15.27219587121160758306368618573, 15.64277916073829568186381517386, 16.432342788292097500386677085466, 17.21194672641510002706168916795, 17.787197284493162728390972209334, 18.6902189332329709940468230873