| L(s) = 1 | − i·2-s − 3-s − 4-s − i·5-s + i·6-s + i·8-s + 9-s − 10-s + 12-s + i·15-s + 16-s + 6·17-s − i·18-s + i·20-s + 4·23-s − i·24-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577·3-s − 0.5·4-s − 0.447i·5-s + 0.408i·6-s + 0.353i·8-s + 0.333·9-s − 0.316·10-s + 0.288·12-s + 0.258i·15-s + 0.250·16-s + 1.45·17-s − 0.235i·18-s + 0.223i·20-s + 0.834·23-s − 0.204i·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.383946285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.383946285\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 10T + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 - 2iT - 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 16iT - 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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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
−8.198135375919331840457778272343, −7.52591359610023929216041127757, −6.73375902614860940196976603363, −5.56760850127032610751125474591, −5.42196958548121132058753421517, −4.39061178178631468494967381001, −3.67494754847133786121739308508, −2.77687199364548383130086485012, −1.59902414252681289599772025813, −0.810313891126520502397480983647,
0.58010209589135118107773433150, 1.84783341576565889441710241662, 3.17229936165954429728502411040, 3.87087845375108420712881376024, 4.83339163629884907695614539457, 5.64863278600151110438589543820, 5.95196049065761391536447184020, 6.99943850519943445645605652065, 7.41234312493628233856416780872, 8.029691840275287450711701158471
