L(s) = 1 | + i·2-s − i·3-s + 4-s + 6-s + 4.47i·7-s + 3i·8-s − 9-s + 3.23·11-s − i·12-s − 3.38i·13-s − 4.47·14-s − 16-s − 2.85i·17-s − i·18-s + 3.23·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.577i·3-s + 0.5·4-s + 0.408·6-s + 1.69i·7-s + 1.06i·8-s − 0.333·9-s + 0.975·11-s − 0.288i·12-s − 0.937i·13-s − 1.19·14-s − 0.250·16-s − 0.692i·17-s − 0.235i·18-s + 0.742·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.174498427\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.174498427\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - iT - 2T^{2} \) |
| 7 | \( 1 - 4.47iT - 7T^{2} \) |
| 11 | \( 1 - 3.23T + 11T^{2} \) |
| 13 | \( 1 + 3.38iT - 13T^{2} \) |
| 17 | \( 1 + 2.85iT - 17T^{2} \) |
| 19 | \( 1 - 3.23T + 19T^{2} \) |
| 23 | \( 1 - 4.47iT - 23T^{2} \) |
| 29 | \( 1 + 4.38T + 29T^{2} \) |
| 31 | \( 1 - 7.23T + 31T^{2} \) |
| 37 | \( 1 - 8.09iT - 37T^{2} \) |
| 41 | \( 1 + 1.38T + 41T^{2} \) |
| 43 | \( 1 - 5.70iT - 43T^{2} \) |
| 47 | \( 1 + 5.23iT - 47T^{2} \) |
| 53 | \( 1 + 1.38iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 0.618T + 61T^{2} \) |
| 67 | \( 1 - 5.23iT - 67T^{2} \) |
| 71 | \( 1 + 0.763T + 71T^{2} \) |
| 73 | \( 1 + 3.09iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 3.52iT - 83T^{2} \) |
| 89 | \( 1 + 7.61T + 89T^{2} \) |
| 97 | \( 1 - 8.85iT - 97T^{2} \) |
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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
−9.205237811719044240414587355571, −8.424137874260243779179281218876, −7.85115591374790055239023364360, −6.98168671972027623108625750796, −6.25437510133196352256652121581, −5.61647136383286495929387550954, −5.01896294771815173120517823183, −3.24116748531025365877659895480, −2.54309790321106650592748592159, −1.42982210817536265918396707598,
0.840701583741897688035134510121, 1.88654195843357651163587658233, 3.24329127143499590381312831835, 4.03959718744722641617174791536, 4.40360998363671062615136146557, 5.93352541655378069526457599140, 6.81553448528601574350266941438, 7.24470851577936723913985263401, 8.337010117163744100699796517049, 9.415447873479258632187152870450