| L(s) = 1 | + i·2-s + i·3-s − 4-s − 6-s − 0.723i·7-s − i·8-s − 9-s + 5.51·11-s − i·12-s + 4.96i·13-s + 0.723·14-s + 16-s − 4.23i·17-s − i·18-s − 4.96·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s − 0.273i·7-s − 0.353i·8-s − 0.333·9-s + 1.66·11-s − 0.288i·12-s + 1.37i·13-s + 0.193·14-s + 0.250·16-s − 1.02i·17-s − 0.235i·18-s − 1.13·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.802056121\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.802056121\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 + iT \) |
| good | 7 | \( 1 + 0.723iT - 7T^{2} \) |
| 11 | \( 1 - 5.51T + 11T^{2} \) |
| 13 | \( 1 - 4.96iT - 13T^{2} \) |
| 17 | \( 1 + 4.23iT - 17T^{2} \) |
| 19 | \( 1 + 4.96T + 19T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 4.23iT - 37T^{2} \) |
| 41 | \( 1 - 1.03T + 41T^{2} \) |
| 43 | \( 1 + 0.961iT - 43T^{2} \) |
| 47 | \( 1 - 7.96iT - 47T^{2} \) |
| 53 | \( 1 + 6.47iT - 53T^{2} \) |
| 59 | \( 1 - 9.02T + 59T^{2} \) |
| 61 | \( 1 - 7.44T + 61T^{2} \) |
| 67 | \( 1 - 11.9iT - 67T^{2} \) |
| 71 | \( 1 + 0.514T + 71T^{2} \) |
| 73 | \( 1 + 0.447iT - 73T^{2} \) |
| 79 | \( 1 - 5.51T + 79T^{2} \) |
| 83 | \( 1 - 6.17iT - 83T^{2} \) |
| 89 | \( 1 - 2.23T + 89T^{2} \) |
| 97 | \( 1 + 2.55iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.817613904672321423827869107538, −8.266734469047876630376910789671, −7.03587606147173777250096966623, −6.70153493248704032212673346890, −6.01657365182289799411533511948, −4.87020475067635571362035501477, −4.28062574055112231648766488315, −3.74715677100648300085133051539, −2.40410188527741793388365319494, −1.03490036169392673534131713959,
0.67511086504351978830107174705, 1.66846619800923977485071334795, 2.57538469926081953333294082133, 3.61273501803460507173150813844, 4.22745965249450570205597122552, 5.38861299737289606246678985603, 6.13920963230802088010641887308, 6.73910145444281789524644164939, 7.79856042557802227803298380097, 8.495396556471943292920900215316
