| L(s) = 1 | + i·2-s + i·3-s − 4-s − 6-s + 3.12i·7-s − i·8-s − 9-s − 3.12·11-s − i·12-s − 2i·13-s − 3.12·14-s + 16-s − 1.12i·17-s − i·18-s − 4·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s + 1.18i·7-s − 0.353i·8-s − 0.333·9-s − 0.941·11-s − 0.288i·12-s − 0.554i·13-s − 0.834·14-s + 0.250·16-s − 0.272i·17-s − 0.235i·18-s − 0.917·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5472300639\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5472300639\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 + iT \) |
| good | 7 | \( 1 - 3.12iT - 7T^{2} \) |
| 11 | \( 1 + 3.12T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 1.12iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 1.12iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 12.2iT - 53T^{2} \) |
| 59 | \( 1 + 2.24T + 59T^{2} \) |
| 61 | \( 1 - 9.12T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 - 4.24iT - 73T^{2} \) |
| 79 | \( 1 + 3.12T + 79T^{2} \) |
| 83 | \( 1 - 13.3iT - 83T^{2} \) |
| 89 | \( 1 - 5.12T + 89T^{2} \) |
| 97 | \( 1 + 16.2iT - 97T^{2} \) |
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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
−8.463450786999017655033719728672, −8.006343135000027705049177656966, −7.01388334887761394724310042241, −6.15601552118731819622433361030, −5.42812621489432081612296715341, −5.03091640169990562188765796575, −4.00806909669439277531778345731, −2.98761961909249122510037690544, −2.15459012697708894074509954153, −0.18146647943804202374705040726,
1.04214622099868023165552605788, 2.04774488281384756475943532410, 2.95625980725005282647279527272, 3.99630004546142160316249411728, 4.57120165832405361994766179720, 5.63133762754498625801228296284, 6.45197132319223057140208347491, 7.36608726262647133465963891538, 7.79750986486942610658853221520, 8.678487290444517110738568073923
