| L(s) = 1 | − i·2-s + i·3-s − 4-s + 6-s − 4.70i·7-s + i·8-s − 9-s + 4.70·11-s − i·12-s − i·13-s − 4.70·14-s + 16-s + 0.701i·17-s + i·18-s + 1.70·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s − 1.77i·7-s + 0.353i·8-s − 0.333·9-s + 1.41·11-s − 0.288i·12-s − 0.277i·13-s − 1.25·14-s + 0.250·16-s + 0.170i·17-s + 0.235i·18-s + 0.390·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.541385529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.541385529\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + iT \) |
| good | 7 | \( 1 + 4.70iT - 7T^{2} \) |
| 11 | \( 1 - 4.70T + 11T^{2} \) |
| 17 | \( 1 - 0.701iT - 17T^{2} \) |
| 19 | \( 1 - 1.70T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 6.40T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 - 1.70iT - 37T^{2} \) |
| 41 | \( 1 + 3.70T + 41T^{2} \) |
| 43 | \( 1 + 11.4iT - 43T^{2} \) |
| 47 | \( 1 + 7iT - 47T^{2} \) |
| 53 | \( 1 - 2.40iT - 53T^{2} \) |
| 59 | \( 1 + 2.70T + 59T^{2} \) |
| 61 | \( 1 - 14.1T + 61T^{2} \) |
| 67 | \( 1 - 6.40iT - 67T^{2} \) |
| 71 | \( 1 + 1.70T + 71T^{2} \) |
| 73 | \( 1 + 12iT - 73T^{2} \) |
| 79 | \( 1 - 5.70T + 79T^{2} \) |
| 83 | \( 1 + 10.7iT - 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 - 2.59iT - 97T^{2} \) |
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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
−9.101140451412929602616259227852, −8.377326998223366147438741686719, −7.29576484209244658744465132287, −6.71787527508120273938962190193, −5.51269871229547537334658996054, −4.51050666248660157111310254164, −3.84142782100584683173524401439, −3.33922028834150082928103229809, −1.70425941825582862535141704260, −0.61160390260752635877283512498,
1.38003294398219440850545399707, 2.51760217844719032227529738067, 3.63020450606631362179428493954, 4.84688342775007845959284980924, 5.64513257792986695834246802822, 6.33725427442236757339706106103, 6.89891286124218098694984882728, 7.925821540282447204791585484211, 8.662689343560125818427540146274, 9.196174645370767530634576728494
