| L(s) = 1 | − i·5-s − 7-s + 3·9-s + 2i·11-s + 2i·13-s + 2·17-s + 4i·19-s − 25-s − 4i·29-s − 4·31-s + i·35-s + 8i·37-s + 2·41-s + 6i·43-s − 3i·45-s + ⋯ |
| L(s) = 1 | − 0.447i·5-s − 0.377·7-s + 9-s + 0.603i·11-s + 0.554i·13-s + 0.485·17-s + 0.917i·19-s − 0.200·25-s − 0.742i·29-s − 0.718·31-s + 0.169i·35-s + 1.31i·37-s + 0.312·41-s + 0.914i·43-s − 0.447i·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.703849624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.703849624\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 4iT - 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 4iT - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 2iT - 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.397709449764440221055686101022, −8.255738821406067026695761339602, −7.63278042005552535079640817554, −6.80948385358197838069408110964, −6.07803381650021556992818869119, −5.04979541485169780751950772946, −4.28874508488671362090151674476, −3.52491755460838951078391393440, −2.15352078325479991227337807495, −1.18156123718087293315766263460,
0.66848935185036218028848394397, 2.08795100772617178767537612804, 3.21449516595576385481954250949, 3.88244353425025010823194155336, 5.02394188561978226573516139176, 5.79596310652948991419745354698, 6.75188847030154227609461940845, 7.28774289953926539584235465854, 8.081889991693258912169349168030, 9.081277249170364609466657343396
