L(s) = 1 | + 2-s + (0.5 − 0.866i)3-s + 4-s + (−0.5 + 0.866i)5-s + (0.5 − 0.866i)6-s + 8-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (0.5 − 0.866i)12-s + (1 + 1.73i)13-s + (0.499 + 0.866i)15-s + 16-s + (−0.499 − 0.866i)18-s + (−0.5 − 0.866i)19-s + (−0.5 + 0.866i)20-s + ⋯ |
L(s) = 1 | + 2-s + (0.5 − 0.866i)3-s + 4-s + (−0.5 + 0.866i)5-s + (0.5 − 0.866i)6-s + 8-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (0.5 − 0.866i)12-s + (1 + 1.73i)13-s + (0.499 + 0.866i)15-s + 16-s + (−0.499 − 0.866i)18-s + (−0.5 − 0.866i)19-s + (−0.5 + 0.866i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.775212163\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.775212163\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + 2T + T^{2} \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−8.640644823119374746115880883853, −7.67209080560245808983818317204, −7.01420059307538445799687140054, −6.57233559091441989553615626593, −6.04601169469397880298624756432, −4.71917968426044402910304257209, −3.97105610271985579430820714388, −3.18916758371789600239158023001, −2.43013537986300548525327794737, −1.49270521567800576496462078567,
1.35533565392483543642070749269, 2.73895024679029356378356042837, 3.51658525456092543167318259752, 4.05974288080245347393360387797, 4.93989378160446447256379283747, 5.51259131153441008494197983211, 6.19408313986781596639960891483, 7.49407314286531536091447711524, 8.106301932817996625733890652855, 8.545033772259034090292289002843