L(s) = 1 | + (0.365 − 0.930i)3-s + (0.988 − 0.149i)5-s + (−0.733 − 0.680i)9-s + (0.733 − 0.680i)11-s + (0.222 + 0.974i)13-s + (0.222 − 0.974i)15-s + (−0.0747 − 0.997i)17-s + (−0.5 + 0.866i)19-s + (−0.0747 + 0.997i)23-s + (0.955 − 0.294i)25-s + (−0.900 + 0.433i)27-s + (−0.900 − 0.433i)29-s + (−0.5 − 0.866i)31-s + (−0.365 − 0.930i)33-s + (0.826 − 0.563i)37-s + ⋯ |
L(s) = 1 | + (0.365 − 0.930i)3-s + (0.988 − 0.149i)5-s + (−0.733 − 0.680i)9-s + (0.733 − 0.680i)11-s + (0.222 + 0.974i)13-s + (0.222 − 0.974i)15-s + (−0.0747 − 0.997i)17-s + (−0.5 + 0.866i)19-s + (−0.0747 + 0.997i)23-s + (0.955 − 0.294i)25-s + (−0.900 + 0.433i)27-s + (−0.900 − 0.433i)29-s + (−0.5 − 0.866i)31-s + (−0.365 − 0.930i)33-s + (0.826 − 0.563i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.283286330 - 0.7873422983i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.283286330 - 0.7873422983i\) |
\(L(1)\) |
\(\approx\) |
\(1.255043262 - 0.4553206887i\) |
\(L(1)\) |
\(\approx\) |
\(1.255043262 - 0.4553206887i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.365 - 0.930i)T \) |
| 5 | \( 1 + (0.988 - 0.149i)T \) |
| 11 | \( 1 + (0.733 - 0.680i)T \) |
| 13 | \( 1 + (0.222 + 0.974i)T \) |
| 17 | \( 1 + (-0.0747 - 0.997i)T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (-0.0747 + 0.997i)T \) |
| 29 | \( 1 + (-0.900 - 0.433i)T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (0.826 - 0.563i)T \) |
| 41 | \( 1 + (-0.623 - 0.781i)T \) |
| 43 | \( 1 + (-0.623 + 0.781i)T \) |
| 47 | \( 1 + (0.955 + 0.294i)T \) |
| 53 | \( 1 + (0.826 + 0.563i)T \) |
| 59 | \( 1 + (-0.988 - 0.149i)T \) |
| 61 | \( 1 + (-0.826 + 0.563i)T \) |
| 67 | \( 1 + (0.5 + 0.866i)T \) |
| 71 | \( 1 + (0.900 - 0.433i)T \) |
| 73 | \( 1 + (-0.955 + 0.294i)T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 + (-0.222 + 0.974i)T \) |
| 89 | \( 1 + (0.733 + 0.680i)T \) |
| 97 | \( 1 - T \) |
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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
−27.15058349194105773739094497027, −26.00445118121614823130514426037, −25.52118962786286117631427786363, −24.61093139214310397315700899666, −23.12273353840799426991366781443, −22.074496883344102155522627148933, −21.651984442707167508825016825087, −20.411118152416788273665102767583, −19.89072016891680475658263013247, −18.40981330476420198513829688337, −17.31904448163467266841279841568, −16.68492820931990266798753493247, −15.20171472273467497933708566616, −14.74009478085074920710540662773, −13.57486102773147672823697996870, −12.58043487930190001547887148197, −10.93346739386452186074780620092, −10.24959705771224047246445954972, −9.26261112082293476589576626572, −8.37549204886477823556418364848, −6.7385725199190002277274388275, −5.5787333075841302199135784448, −4.463969131902142705518550159505, −3.16400976734237504100032162868, −1.89812381673755897376992765892,
1.32867384581257188897939066444, 2.35189695033081414657809997330, 3.83545034719545207623949920725, 5.65402337089482940319895398411, 6.428731610120600870798632646465, 7.56889014357312765144757580905, 8.92408308850962049445107519108, 9.510445026152584152524606484270, 11.18823158693894911018463292389, 12.11641160363198574024492239283, 13.3727034811627496881634844043, 13.88504240862085164192958779610, 14.79112434024449803337263491729, 16.49520694270274627291486329161, 17.19588294732533609677519150373, 18.35958865570711738094051321613, 18.95378942477095782460146782300, 20.1221118980440599056891645217, 21.04935856554787919619805659447, 22.001115438110381385835962885348, 23.17898021630597669787058655259, 24.22037263667219383168673662536, 24.91970486459553111270381330628, 25.6569720988163374216323530397, 26.58908156382296470034717516337