L(s) = 1 | + (1 + 1.41i)3-s − 1.41i·5-s + (2 + 1.73i)7-s + (−1.00 + 2.82i)9-s + 2.44·11-s + 2·13-s + (2.00 − 1.41i)15-s − 7.34·17-s + 4·19-s + (−0.449 + 4.56i)21-s + 1.41i·23-s + 2.99·25-s + (−5.00 + 1.41i)27-s + 4.89·29-s + 6.92i·31-s + ⋯ |
L(s) = 1 | + (0.577 + 0.816i)3-s − 0.632i·5-s + (0.755 + 0.654i)7-s + (−0.333 + 0.942i)9-s + 0.738·11-s + 0.554·13-s + (0.516 − 0.365i)15-s − 1.78·17-s + 0.917·19-s + (−0.0980 + 0.995i)21-s + 0.294i·23-s + 0.599·25-s + (−0.962 + 0.272i)27-s + 0.909·29-s + 1.24i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.582 - 0.812i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.582 - 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.77457 + 0.911438i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.77457 + 0.911438i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1 - 1.41i)T \) |
| 7 | \( 1 + (-2 - 1.73i)T \) |
good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 11 | \( 1 - 2.44T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 7.34T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 1.41iT - 23T^{2} \) |
| 29 | \( 1 - 4.89T + 29T^{2} \) |
| 31 | \( 1 - 6.92iT - 31T^{2} \) |
| 37 | \( 1 + 10.3iT - 37T^{2} \) |
| 41 | \( 1 - 2.44T + 41T^{2} \) |
| 43 | \( 1 - 3.46iT - 43T^{2} \) |
| 47 | \( 1 + 4.89T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 5.65iT - 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 3.46iT - 67T^{2} \) |
| 71 | \( 1 - 1.41iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 11.3iT - 83T^{2} \) |
| 89 | \( 1 - 7.34T + 89T^{2} \) |
| 97 | \( 1 + 13.8iT - 97T^{2} \) |
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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
−10.73307730591929299188336634433, −9.487440448183907426520442882120, −8.822818831255794580235909779393, −8.500453371922550874692180595537, −7.26757041922376147309717620829, −5.96717078742222337041036617604, −4.90331169891939580757818045993, −4.30699378474233078927342964650, −3.00053355294813182512828536706, −1.66008500783721727014383367316,
1.17661095986436895765425986366, 2.45631059065029326884394605721, 3.64747687004886901258438538912, 4.70502769867762862235287280663, 6.35727032650156535385128502522, 6.78396835496879474492638133681, 7.74483686259312727871397010446, 8.529416769892170501949845477809, 9.339776203453619994220502500902, 10.48551845970547180575445614003