L(s) = 1 | − 3.14i·5-s − i·7-s + 2.04·11-s + 2.44·13-s − 3.14i·17-s − 4.44i·19-s + 2.82·23-s − 4.89·25-s − 0.778i·29-s − 0.449i·31-s − 3.14·35-s − 7·37-s − 2.51i·41-s − 4.34i·43-s + 7.38·47-s + ⋯ |
L(s) = 1 | − 1.40i·5-s − 0.377i·7-s + 0.618·11-s + 0.679·13-s − 0.763i·17-s − 1.02i·19-s + 0.589·23-s − 0.979·25-s − 0.144i·29-s − 0.0807i·31-s − 0.531·35-s − 1.15·37-s − 0.392i·41-s − 0.663i·43-s + 1.07·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.931235003\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.931235003\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 + 3.14iT - 5T^{2} \) |
| 11 | \( 1 - 2.04T + 11T^{2} \) |
| 13 | \( 1 - 2.44T + 13T^{2} \) |
| 17 | \( 1 + 3.14iT - 17T^{2} \) |
| 19 | \( 1 + 4.44iT - 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 + 0.778iT - 29T^{2} \) |
| 31 | \( 1 + 0.449iT - 31T^{2} \) |
| 37 | \( 1 + 7T + 37T^{2} \) |
| 41 | \( 1 + 2.51iT - 41T^{2} \) |
| 43 | \( 1 + 4.34iT - 43T^{2} \) |
| 47 | \( 1 - 7.38T + 47T^{2} \) |
| 53 | \( 1 + 6.29iT - 53T^{2} \) |
| 59 | \( 1 - 5.83T + 59T^{2} \) |
| 61 | \( 1 + 5.34T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 79 | \( 1 - 13.4iT - 79T^{2} \) |
| 83 | \( 1 - 6.75T + 83T^{2} \) |
| 89 | \( 1 - 14.1iT - 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−7.933133866406348666958008813935, −7.02428608087414218370263609475, −6.52304840510245321695218111674, −5.37155804526808275238023989559, −5.04686570271960202962683837518, −4.17046117564592244188250991410, −3.53591086787433434216554475521, −2.32560024535311574376018493353, −1.22124955038903024331015413269, −0.54380689866302118512548216040,
1.33666832799035833835007887596, 2.25641014701022329842338303078, 3.26167052563177931259275649550, 3.66943281249809802027528379249, 4.65230653866770991886957885538, 5.86923660362301260510379645309, 6.10859546427307442350465165493, 6.93783244649565450184391588844, 7.47371968536023046365922874642, 8.397082085384362547733609054713