L(s) = 1 | + (1 − i)2-s − i·4-s + (−0.707 − 0.707i)7-s − 1.41i·11-s − 1.41·14-s + 16-s + (−1.41 − 1.41i)22-s + (−1 − i)23-s + (−0.707 + 0.707i)28-s + 1.41·29-s + (1 − i)32-s + (1.41 + 1.41i)37-s + (−1.41 + 1.41i)43-s − 1.41·44-s − 2·46-s + ⋯ |
L(s) = 1 | + (1 − i)2-s − i·4-s + (−0.707 − 0.707i)7-s − 1.41i·11-s − 1.41·14-s + 16-s + (−1.41 − 1.41i)22-s + (−1 − i)23-s + (−0.707 + 0.707i)28-s + 1.41·29-s + (1 − i)32-s + (1.41 + 1.41i)37-s + (−1.41 + 1.41i)43-s − 1.41·44-s − 2·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.751838313\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.751838313\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
good | 2 | \( 1 + (-1 + i)T - iT^{2} \) |
| 11 | \( 1 + 1.41iT - T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (1 + i)T + iT^{2} \) |
| 29 | \( 1 - 1.41T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (1.41 - 1.41i)T - iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (1 + i)T + iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 - 1.41iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 - 2iT - T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−9.889281528924600794794481440043, −8.466219262566417476036316629381, −7.976276606161320309392002487064, −6.56215842585024997409073439737, −6.09312313545987489851557351538, −4.92000633166331703893434148320, −4.15706526693404224822204476378, −3.28527492964716103587803528489, −2.66671426068583057859247343096, −1.09574911540786564503374128921,
1.99853543848525930454647359952, 3.26212981803990160417194471414, 4.24405567623656101125730488152, 4.99473716344798591472882486957, 5.87062846748685346960231179231, 6.46141937531656666784095877689, 7.29050977855640185738331891894, 7.914334908967090917684493587885, 9.045671556612517321104457729427, 9.815058115810703438957590670150