L(s) = 1 | + 1.41i·2-s − 2.00·4-s + (2.23 + 0.158i)5-s − 2.82i·8-s + 3·9-s + (−0.224 + 3.15i)10-s − 4.01·13-s + 4.00·16-s + 4.90·17-s + 4.24i·18-s + (−4.46 − 0.317i)20-s + (4.94 + 0.707i)25-s − 5.67i·26-s + 9.89·29-s + 5.65i·32-s + ⋯ |
L(s) = 1 | + 0.999i·2-s − 1.00·4-s + (0.997 + 0.0708i)5-s − 1.00i·8-s + 9-s + (−0.0708 + 0.997i)10-s − 1.11·13-s + 1.00·16-s + 1.19·17-s + 0.999i·18-s + (−0.997 − 0.0708i)20-s + (0.989 + 0.141i)25-s − 1.11i·26-s + 1.83·29-s + 1.00i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.226 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.226 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.43638 + 1.14107i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.43638 + 1.14107i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41iT \) |
| 5 | \( 1 + (-2.23 - 0.158i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 3T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 4.01T + 13T^{2} \) |
| 17 | \( 1 - 4.90T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 9.89T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 7.07iT - 37T^{2} \) |
| 41 | \( 1 + 12.3iT - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 14iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 7.25iT - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 18.6iT - 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−10.06438662731779831054982649145, −9.385088850295493746826603219247, −8.452240315933521957831635007920, −7.43206757452755982191144057513, −6.87925305179000431581758240612, −5.92255696881851980917523226776, −5.11113559393278825025008289885, −4.30455701381110122512836155141, −2.85913017237234035586513587731, −1.24222474089827617391807364323,
1.11177865793488242244567892667, 2.20813030549432332577024390540, 3.23192940031634471032156033311, 4.56025266238916972212059351988, 5.16481355226088513716959457058, 6.28412344136081695375262608101, 7.40202496490079918058706116787, 8.355843887562121630228923343964, 9.405929126246602088173773450069, 10.02431234144885671502755118868