L(s) = 1 | + 2.44i·2-s + i·3-s − 3.95·4-s − 2.44·6-s − 3.44i·7-s − 4.77i·8-s − 9-s − 3.26·11-s − 3.95i·12-s + 3.23i·13-s + 8.39·14-s + 3.73·16-s − 5.05i·17-s − 2.44i·18-s + 3.08·19-s + ⋯ |
L(s) = 1 | + 1.72i·2-s + 0.577i·3-s − 1.97·4-s − 0.996·6-s − 1.30i·7-s − 1.68i·8-s − 0.333·9-s − 0.984·11-s − 1.14i·12-s + 0.896i·13-s + 2.24·14-s + 0.932·16-s − 1.22i·17-s − 0.575i·18-s + 0.706·19-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
\(L(1)\) |
\(\approx\) |
\(1.239367976\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.239367976\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.44iT - 2T^{2} \) |
| 7 | \( 1 + 3.44iT - 7T^{2} \) |
| 11 | \( 1 + 3.26T + 11T^{2} \) |
| 13 | \( 1 - 3.23iT - 13T^{2} \) |
| 17 | \( 1 + 5.05iT - 17T^{2} \) |
| 19 | \( 1 - 3.08T + 19T^{2} \) |
| 23 | \( 1 + 1.54iT - 23T^{2} \) |
| 29 | \( 1 - 3.12T + 29T^{2} \) |
| 31 | \( 1 - 7.44T + 31T^{2} \) |
| 37 | \( 1 + 5.75iT - 37T^{2} \) |
| 41 | \( 1 - 5.41T + 41T^{2} \) |
| 43 | \( 1 + 2.53iT - 43T^{2} \) |
| 47 | \( 1 - 7.07iT - 47T^{2} \) |
| 53 | \( 1 - 10.1iT - 53T^{2} \) |
| 59 | \( 1 + 1.73T + 59T^{2} \) |
| 61 | \( 1 - 7.83T + 61T^{2} \) |
| 67 | \( 1 + 1.84iT - 67T^{2} \) |
| 71 | \( 1 + 0.713T + 71T^{2} \) |
| 73 | \( 1 - 1.88iT - 73T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 + 3.95iT - 83T^{2} \) |
| 89 | \( 1 + 8.53T + 89T^{2} \) |
| 97 | \( 1 + 10.6iT - 97T^{2} \) |
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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
−9.304964049830543979072292619439, −8.432166257793251119050458556204, −7.58299975104818708513654837506, −7.20544932995822755140873449016, −6.35846930046388057686003638014, −5.43639084434964662865717340234, −4.63004775721843002107909956524, −4.18394225843439430559372590759, −2.82810229041074814123678861961, −0.59496310387755354301962951604,
0.958010091568888775409772404830, 2.16632413346218782968222287417, 2.76094119896526425079334901269, 3.57114128708325140675737231183, 4.92306738621589855353480892621, 5.53217451409616984392081791132, 6.49361254425967125318777380614, 7.994466609706461968932205256989, 8.308780219708193493643493602304, 9.195970645319637469712194752383