L(s) = 1 | + (−0.858 − 1.50i)3-s + (−3.42 − 3.42i)7-s + (−1.52 + 2.58i)9-s − 4.41i·11-s + (−2.37 + 2.37i)13-s + (−2.76 + 2.76i)17-s + 5.73i·19-s + (−2.21 + 8.08i)21-s + (−4.22 − 4.22i)23-s + (5.19 + 0.0829i)27-s + 9.67·29-s − 0.881·31-s + (−6.64 + 3.78i)33-s + (4.89 + 4.89i)37-s + (5.61 + 1.53i)39-s + ⋯ |
L(s) = 1 | + (−0.495 − 0.868i)3-s + (−1.29 − 1.29i)7-s + (−0.509 + 0.860i)9-s − 1.33i·11-s + (−0.659 + 0.659i)13-s + (−0.669 + 0.669i)17-s + 1.31i·19-s + (−0.483 + 1.76i)21-s + (−0.880 − 0.880i)23-s + (0.999 + 0.0159i)27-s + 1.79·29-s − 0.158·31-s + (−1.15 + 0.659i)33-s + (0.804 + 0.804i)37-s + (0.899 + 0.246i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.263 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2054003889\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2054003889\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.858 + 1.50i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (3.42 + 3.42i)T + 7iT^{2} \) |
| 11 | \( 1 + 4.41iT - 11T^{2} \) |
| 13 | \( 1 + (2.37 - 2.37i)T - 13iT^{2} \) |
| 17 | \( 1 + (2.76 - 2.76i)T - 17iT^{2} \) |
| 19 | \( 1 - 5.73iT - 19T^{2} \) |
| 23 | \( 1 + (4.22 + 4.22i)T + 23iT^{2} \) |
| 29 | \( 1 - 9.67T + 29T^{2} \) |
| 31 | \( 1 + 0.881T + 31T^{2} \) |
| 37 | \( 1 + (-4.89 - 4.89i)T + 37iT^{2} \) |
| 41 | \( 1 + 4.74iT - 41T^{2} \) |
| 43 | \( 1 + (-2.35 + 2.35i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.71 + 1.71i)T - 47iT^{2} \) |
| 53 | \( 1 + (-0.0871 - 0.0871i)T + 53iT^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 + 8.20T + 61T^{2} \) |
| 67 | \( 1 + (-5.36 - 5.36i)T + 67iT^{2} \) |
| 71 | \( 1 - 5.14iT - 71T^{2} \) |
| 73 | \( 1 + (-0.0238 + 0.0238i)T - 73iT^{2} \) |
| 79 | \( 1 - 10.1iT - 79T^{2} \) |
| 83 | \( 1 + (-3.11 - 3.11i)T + 83iT^{2} \) |
| 89 | \( 1 - 2.72T + 89T^{2} \) |
| 97 | \( 1 + (4.70 + 4.70i)T + 97iT^{2} \) |
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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
−9.866459649402853412762318002299, −8.656692102069721174244996669286, −7.975255939695191771155162298867, −7.06782855300475152833885531431, −6.33093660227736970745229105280, −6.01530987540823519155751656845, −4.54050753865712085543975868577, −3.65813776730567081872257793829, −2.53884125985461633656913577821, −1.05656960891236449195421910349,
0.098747754537518333935636874634, 2.46977568627172791786626853540, 3.08689819079610378104348387435, 4.47688072830520027811677966447, 5.04997698203425528427435482743, 6.03975504475906491893135335234, 6.64298322456763641409081696494, 7.65510886197964660296672836789, 8.950060884433964446580947371720, 9.453894727690682101544380655772