L(s) = 1 | + (−0.433 − 0.900i)2-s + (−0.781 + 0.623i)3-s + (−0.623 + 0.781i)4-s + (0.900 − 0.433i)5-s + (0.900 + 0.433i)6-s + (0.623 + 0.781i)7-s + (0.974 + 0.222i)8-s + (0.222 − 0.974i)9-s + (−0.781 − 0.623i)10-s + (0.974 − 0.222i)11-s − i·12-s + (0.222 + 0.974i)13-s + (0.433 − 0.900i)14-s + (−0.433 + 0.900i)15-s + (−0.222 − 0.974i)16-s + i·17-s + ⋯ |
L(s) = 1 | + (−0.433 − 0.900i)2-s + (−0.781 + 0.623i)3-s + (−0.623 + 0.781i)4-s + (0.900 − 0.433i)5-s + (0.900 + 0.433i)6-s + (0.623 + 0.781i)7-s + (0.974 + 0.222i)8-s + (0.222 − 0.974i)9-s + (−0.781 − 0.623i)10-s + (0.974 − 0.222i)11-s − i·12-s + (0.222 + 0.974i)13-s + (0.433 − 0.900i)14-s + (−0.433 + 0.900i)15-s + (−0.222 − 0.974i)16-s + i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.995 - 0.0915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.995 - 0.0915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.036806000 - 0.04756654195i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.036806000 - 0.04756654195i\) |
\(L(1)\) |
\(\approx\) |
\(0.8568338239 - 0.1013611466i\) |
\(L(1)\) |
\(\approx\) |
\(0.8568338239 - 0.1013611466i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 29 | \( 1 \) |
good | 2 | \( 1 + (-0.433 - 0.900i)T \) |
| 3 | \( 1 + (-0.781 + 0.623i)T \) |
| 5 | \( 1 + (0.900 - 0.433i)T \) |
| 7 | \( 1 + (0.623 + 0.781i)T \) |
| 11 | \( 1 + (0.974 - 0.222i)T \) |
| 13 | \( 1 + (0.222 + 0.974i)T \) |
| 17 | \( 1 + iT \) |
| 19 | \( 1 + (0.781 + 0.623i)T \) |
| 23 | \( 1 + (-0.900 - 0.433i)T \) |
| 31 | \( 1 + (-0.433 - 0.900i)T \) |
| 37 | \( 1 + (0.974 + 0.222i)T \) |
| 41 | \( 1 - iT \) |
| 43 | \( 1 + (0.433 - 0.900i)T \) |
| 47 | \( 1 + (-0.974 + 0.222i)T \) |
| 53 | \( 1 + (-0.900 + 0.433i)T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + (-0.781 + 0.623i)T \) |
| 67 | \( 1 + (0.222 - 0.974i)T \) |
| 71 | \( 1 + (0.222 + 0.974i)T \) |
| 73 | \( 1 + (-0.433 + 0.900i)T \) |
| 79 | \( 1 + (-0.974 - 0.222i)T \) |
| 83 | \( 1 + (0.623 - 0.781i)T \) |
| 89 | \( 1 + (-0.433 - 0.900i)T \) |
| 97 | \( 1 + (-0.781 - 0.623i)T \) |
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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
−36.6768024750537109595614462980, −35.68088718644992583502312664157, −34.45299824367336704189971217567, −33.50346117751801689106783340171, −32.701769640861338063607246612684, −30.480061552927352693110962892897, −29.49190165863024297267883300812, −28.05713835018505960569956812958, −26.92760552938856117912223062680, −25.33963623871959342561630656119, −24.49370962785625693134520127697, −23.132056465446898692182859060137, −22.14876747301269972330489422089, −19.88819554539911183253875080922, −18.0067324705171953152122664261, −17.70197294184075725435759899942, −16.361443465830924156368184224211, −14.398275635453950716661499121741, −13.38303300051010020536719005665, −11.15222924562921701289161606038, −9.76231287548769352824883963993, −7.646467688411418059686950519492, −6.49438789284978729864913647264, −5.100224694738397674018041688564, −1.206309939527681689256645327432,
1.624482192777851356915596144287, 4.2129384210276921064335887856, 5.88751288896548736915049419477, 8.77086558335585713097009621461, 9.81658537106823440144192915625, 11.34906711451598779326578306977, 12.375304923821431352259963061217, 14.27105756673622993132773140345, 16.47835026237073442330052541516, 17.447899638051577674459894302521, 18.59836050650690781555393859297, 20.52640504900480856883949471653, 21.57150718591431718674530546422, 22.22169229728953273153179738117, 24.2306076385507521874195727213, 25.91899522446749659210779286520, 27.356328726803946972941983802980, 28.30981461887925118996928413111, 29.0777791714324372657860602526, 30.45736021203227074546160197293, 32.007523844569122851497968043323, 33.276553473324365551164334191734, 34.62698441998043606967371491405, 35.83210771044771497461821649231, 37.37645670041370420854384355795