| L(s) = 1 | + i·2-s − 4-s + (−1 + 2i)5-s − i·8-s + (−2 − i)10-s + 6·11-s − 3i·13-s + 16-s − i·17-s + 7·19-s + (1 − 2i)20-s + 6i·22-s − 8i·23-s + (−3 − 4i)25-s + 3·26-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (−0.447 + 0.894i)5-s − 0.353i·8-s + (−0.632 − 0.316i)10-s + 1.80·11-s − 0.832i·13-s + 0.250·16-s − 0.242i·17-s + 1.60·19-s + (0.223 − 0.447i)20-s + 1.27i·22-s − 1.66i·23-s + (−0.600 − 0.800i)25-s + 0.588·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.674603852\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.674603852\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + (1 - 2i)T \) |
| 17 | \( 1 + iT \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 13 | \( 1 + 3iT - 13T^{2} \) |
| 19 | \( 1 - 7T + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 3iT - 47T^{2} \) |
| 53 | \( 1 - 9iT - 53T^{2} \) |
| 59 | \( 1 - 5T + 59T^{2} \) |
| 61 | \( 1 + 3T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 - 15T + 71T^{2} \) |
| 73 | \( 1 - 11iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + T + 89T^{2} \) |
| 97 | \( 1 + 9iT - 97T^{2} \) |
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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
−9.574940155004068704652570395045, −8.662936361103871716989379782131, −7.897249268489703393040747064930, −7.06099684890190158545618563299, −6.53485010185974203651832092941, −5.70851440163146030420426718058, −4.55442315016337062262688066503, −3.69654081797907632355027308737, −2.80386370735498202050262349264, −0.965179728605209458664857058859,
1.00277490902474422367687790161, 1.83488451410822797570573424046, 3.59539752686898712672569066907, 3.92914095735638091347354450585, 5.01603686263212382262767009302, 5.85587332066387977612109842558, 7.04560388598206220427067239515, 7.78848156809140334927639841291, 8.924553204268365725342313851439, 9.262883563001979810427814166479
