L(s) = 1 | − i·2-s + (−0.707 + 0.707i)3-s − 4-s + (−0.325 + 2.21i)5-s + (0.707 + 0.707i)6-s + (0.597 − 0.597i)7-s + i·8-s − 1.00i·9-s + (2.21 + 0.325i)10-s + 6.22i·11-s + (0.707 − 0.707i)12-s − 0.986i·13-s + (−0.597 − 0.597i)14-s + (−1.33 − 1.79i)15-s + 16-s − 5.73·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (−0.408 + 0.408i)3-s − 0.5·4-s + (−0.145 + 0.989i)5-s + (0.288 + 0.288i)6-s + (0.225 − 0.225i)7-s + 0.353i·8-s − 0.333i·9-s + (0.699 + 0.102i)10-s + 1.87i·11-s + (0.204 − 0.204i)12-s − 0.273i·13-s + (−0.159 − 0.159i)14-s + (−0.344 − 0.463i)15-s + 0.250·16-s − 1.39·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.802 - 0.596i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.802 - 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4610899588\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4610899588\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (0.325 - 2.21i)T \) |
| 37 | \( 1 + (6.02 - 0.865i)T \) |
good | 7 | \( 1 + (-0.597 + 0.597i)T - 7iT^{2} \) |
| 11 | \( 1 - 6.22iT - 11T^{2} \) |
| 13 | \( 1 + 0.986iT - 13T^{2} \) |
| 17 | \( 1 + 5.73T + 17T^{2} \) |
| 19 | \( 1 + (-1.99 + 1.99i)T - 19iT^{2} \) |
| 23 | \( 1 + 1.91iT - 23T^{2} \) |
| 29 | \( 1 + (3.12 + 3.12i)T + 29iT^{2} \) |
| 31 | \( 1 + (0.922 - 0.922i)T - 31iT^{2} \) |
| 41 | \( 1 + 1.55iT - 41T^{2} \) |
| 43 | \( 1 + 6.23iT - 43T^{2} \) |
| 47 | \( 1 + (2.10 - 2.10i)T - 47iT^{2} \) |
| 53 | \( 1 + (-9.01 - 9.01i)T + 53iT^{2} \) |
| 59 | \( 1 + (5.56 - 5.56i)T - 59iT^{2} \) |
| 61 | \( 1 + (1.20 - 1.20i)T - 61iT^{2} \) |
| 67 | \( 1 + (9.76 + 9.76i)T + 67iT^{2} \) |
| 71 | \( 1 + 14.9T + 71T^{2} \) |
| 73 | \( 1 + (8.62 - 8.62i)T - 73iT^{2} \) |
| 79 | \( 1 + (4.54 - 4.54i)T - 79iT^{2} \) |
| 83 | \( 1 + (7.29 + 7.29i)T + 83iT^{2} \) |
| 89 | \( 1 + (-3.84 - 3.84i)T + 89iT^{2} \) |
| 97 | \( 1 - 12.5T + 97T^{2} \) |
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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
−10.44877193314338051126613877249, −9.591145301648128894940821997527, −8.828876083864317855002637031385, −7.44714996398614050243506093065, −7.02446259603098889467051236464, −5.86423318131108124657704497428, −4.63984705997157603324537562997, −4.16661056390306667755415563634, −2.87322823573568726912256599244, −1.88895607627341428896544919611,
0.21181761197620940707718021526, 1.60323535246489171540196375757, 3.40299597718114233987193354984, 4.52698418649094422985776376969, 5.46821850635883545926776743556, 5.97452841803068657086805119763, 6.97893196111099990259027662734, 7.935382547364537092749598962479, 8.707456566215824680964343459909, 9.024859815043676926179309788847