L(s) = 1 | + 3.24i·3-s − 0.858i·7-s − 7.49·9-s + 6.20·11-s − 1.41i·13-s − 3.93i·17-s − 3.82·19-s + 2.78·21-s − 3.06i·23-s − 14.5i·27-s + 8.48·29-s + 1.13·31-s + 20.1i·33-s + 8.49i·37-s + 4.58·39-s + ⋯ |
L(s) = 1 | + 1.87i·3-s − 0.324i·7-s − 2.49·9-s + 1.87·11-s − 0.392i·13-s − 0.953i·17-s − 0.877·19-s + 0.607·21-s − 0.639i·23-s − 2.80i·27-s + 1.57·29-s + 0.203·31-s + 3.50i·33-s + 1.39i·37-s + 0.734·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.974790308\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.974790308\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 3 | \( 1 - 3.24iT - 3T^{2} \) |
| 7 | \( 1 + 0.858iT - 7T^{2} \) |
| 11 | \( 1 - 6.20T + 11T^{2} \) |
| 13 | \( 1 + 1.41iT - 13T^{2} \) |
| 17 | \( 1 + 3.93iT - 17T^{2} \) |
| 19 | \( 1 + 3.82T + 19T^{2} \) |
| 23 | \( 1 + 3.06iT - 23T^{2} \) |
| 29 | \( 1 - 8.48T + 29T^{2} \) |
| 31 | \( 1 - 1.13T + 31T^{2} \) |
| 37 | \( 1 - 8.49iT - 37T^{2} \) |
| 43 | \( 1 + 5.34iT - 43T^{2} \) |
| 47 | \( 1 + 6.65iT - 47T^{2} \) |
| 53 | \( 1 + 6.41iT - 53T^{2} \) |
| 59 | \( 1 - 3.06T + 59T^{2} \) |
| 61 | \( 1 - 7.41T + 61T^{2} \) |
| 67 | \( 1 - 1.79iT - 67T^{2} \) |
| 71 | \( 1 + 3.02T + 71T^{2} \) |
| 73 | \( 1 + 0.632iT - 73T^{2} \) |
| 79 | \( 1 - 14.3T + 79T^{2} \) |
| 83 | \( 1 - 8.76iT - 83T^{2} \) |
| 89 | \( 1 - 7.89T + 89T^{2} \) |
| 97 | \( 1 - 9.86iT - 97T^{2} \) |
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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
−8.716795026881668273685919150788, −8.237525198381083670886484650990, −6.81203864206503250776208143621, −6.38463214035087349630861486760, −5.30374247828748539327403738505, −4.66233989158606377212149769204, −4.02616370614920737556044129258, −3.43675698262255137316389845298, −2.46173986802052907594294309606, −0.74489697905913696173078571311,
0.925453176964695748369816735845, 1.67108024071376353396434373532, 2.43092277511463363102581008465, 3.55286021176547737317131479639, 4.47061705737293372137325440476, 5.81296172464417125806273525468, 6.27061038341598966188331386962, 6.74603245619807885496443075875, 7.46257968241262147575232299926, 8.274614000944226937361056173605