L(s) = 1 | + i·2-s − i·3-s − 4-s + 6-s − 0.533i·7-s − i·8-s − 9-s − 1.43·11-s + i·12-s − 6.57i·13-s + 0.533·14-s + 16-s − 0.958i·17-s − i·18-s − 0.212·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s − 0.201i·7-s − 0.353i·8-s − 0.333·9-s − 0.432·11-s + 0.288i·12-s − 1.82i·13-s + 0.142·14-s + 0.250·16-s − 0.232i·17-s − 0.235i·18-s − 0.0488·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1324486800\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1324486800\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 7 | \( 1 + 0.533iT - 7T^{2} \) |
| 11 | \( 1 + 1.43T + 11T^{2} \) |
| 13 | \( 1 + 6.57iT - 13T^{2} \) |
| 17 | \( 1 + 0.958iT - 17T^{2} \) |
| 19 | \( 1 + 0.212T + 19T^{2} \) |
| 23 | \( 1 + 3.76iT - 23T^{2} \) |
| 29 | \( 1 + 6.19T + 29T^{2} \) |
| 31 | \( 1 + 2.33T + 31T^{2} \) |
| 37 | \( 1 - 4.06iT - 37T^{2} \) |
| 41 | \( 1 + 7.94T + 41T^{2} \) |
| 43 | \( 1 - 11.3iT - 43T^{2} \) |
| 47 | \( 1 - 10.1iT - 47T^{2} \) |
| 53 | \( 1 - 3.23iT - 53T^{2} \) |
| 59 | \( 1 + 7.52T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 + 6.91iT - 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 - 13.9iT - 73T^{2} \) |
| 79 | \( 1 + 15.5T + 79T^{2} \) |
| 83 | \( 1 + 16.3iT - 83T^{2} \) |
| 89 | \( 1 + 5.62T + 89T^{2} \) |
| 97 | \( 1 + 5.56iT - 97T^{2} \) |
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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
−7.990029138845986977007852134010, −7.50647679981004974476426784721, −6.72972298233009220249541506670, −5.90309392268583137521924923618, −5.36245875173617713927156583640, −4.51805200700398386244275409367, −3.36930613789030895565680915405, −2.61670221503362409469691949205, −1.18131703825804560583152672820, −0.04027438930330953810016800009,
1.71552123569408679336538730336, 2.37451573293392408961979710484, 3.71458660913321820402697725232, 3.95888377509033898778124210343, 5.10992728337711569513612064845, 5.57440538881152588443287857132, 6.72103075585304244009171785532, 7.40183346638906597358779898758, 8.510310816406402213690555653623, 8.969748265401595008818854447821