L(s) = 1 | − 3.41·7-s − 2.58·11-s − 3.41i·13-s + 1.17·17-s − 4.82i·23-s − 6i·29-s − 6.48i·31-s + 9.07i·37-s + 11.0i·41-s − 6.82·43-s − 5.65i·47-s + 4.65·49-s − 1.17·53-s + 6.58·59-s + 12.8·61-s + ⋯ |
L(s) = 1 | − 1.29·7-s − 0.779·11-s − 0.946i·13-s + 0.284·17-s − 1.00i·23-s − 1.11i·29-s − 1.16i·31-s + 1.49i·37-s + 1.72i·41-s − 1.04·43-s − 0.825i·47-s + 0.665·49-s − 0.160·53-s + 0.857·59-s + 1.64·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4031635136\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4031635136\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 3.41T + 7T^{2} \) |
| 11 | \( 1 + 2.58T + 11T^{2} \) |
| 13 | \( 1 + 3.41iT - 13T^{2} \) |
| 17 | \( 1 - 1.17T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 4.82iT - 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 + 6.48iT - 31T^{2} \) |
| 37 | \( 1 - 9.07iT - 37T^{2} \) |
| 41 | \( 1 - 11.0iT - 41T^{2} \) |
| 43 | \( 1 + 6.82T + 43T^{2} \) |
| 47 | \( 1 + 5.65iT - 47T^{2} \) |
| 53 | \( 1 + 1.17T + 53T^{2} \) |
| 59 | \( 1 - 6.58T + 59T^{2} \) |
| 61 | \( 1 - 12.8T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 + 10.4iT - 73T^{2} \) |
| 79 | \( 1 - 14.4iT - 79T^{2} \) |
| 83 | \( 1 + 9.17iT - 83T^{2} \) |
| 89 | \( 1 - 4.24iT - 89T^{2} \) |
| 97 | \( 1 + 2.48iT - 97T^{2} \) |
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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
−8.090068835700690227173240431967, −7.53399343130081352573984404796, −6.52154445500981819886992963709, −6.21168610103258061444810050343, −5.35531169134511589086403493593, −4.62185788831199352584417306270, −3.63724946069284620817362666859, −2.96242257480360393066443252093, −2.33132675102673531319378318618, −0.815334436864644077930217430145,
0.12371436441068467045418841298, 1.53157596906930454494637877286, 2.53848758468994478253275343742, 3.37944928818316344147309269836, 3.90854125550556522160460855502, 5.02614859713352584923845810512, 5.58827842360510410042799300938, 6.37796826255552673629989829318, 7.12272522169466146391245002730, 7.43624052044888195397069794010