L(s) = 1 | + 0.475i·2-s + 0.345·3-s + 1.77·4-s + 0.161i·5-s + 0.164i·6-s + 4.07i·7-s + 1.79i·8-s − 2.88·9-s − 0.0768·10-s + 3.77i·11-s + 0.613·12-s + (−3.57 − 0.487i)13-s − 1.94·14-s + 0.0558i·15-s + 2.69·16-s + 1.83·17-s + ⋯ |
L(s) = 1 | + 0.336i·2-s + 0.199·3-s + 0.886·4-s + 0.0722i·5-s + 0.0671i·6-s + 1.54i·7-s + 0.634i·8-s − 0.960·9-s − 0.0243·10-s + 1.13i·11-s + 0.176·12-s + (−0.990 − 0.135i)13-s − 0.518·14-s + 0.0144i·15-s + 0.673·16-s + 0.444·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.135 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.135 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.22827 + 1.07203i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.22827 + 1.07203i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 + (3.57 + 0.487i)T \) |
| 31 | \( 1 + iT \) |
good | 2 | \( 1 - 0.475iT - 2T^{2} \) |
| 3 | \( 1 - 0.345T + 3T^{2} \) |
| 5 | \( 1 - 0.161iT - 5T^{2} \) |
| 7 | \( 1 - 4.07iT - 7T^{2} \) |
| 11 | \( 1 - 3.77iT - 11T^{2} \) |
| 17 | \( 1 - 1.83T + 17T^{2} \) |
| 19 | \( 1 + 6.07iT - 19T^{2} \) |
| 23 | \( 1 - 8.58T + 23T^{2} \) |
| 29 | \( 1 + 5.21T + 29T^{2} \) |
| 37 | \( 1 + 11.7iT - 37T^{2} \) |
| 41 | \( 1 - 4.76iT - 41T^{2} \) |
| 43 | \( 1 - 6.24T + 43T^{2} \) |
| 47 | \( 1 + 4.70iT - 47T^{2} \) |
| 53 | \( 1 - 6.16T + 53T^{2} \) |
| 59 | \( 1 - 7.98iT - 59T^{2} \) |
| 61 | \( 1 + 4.52T + 61T^{2} \) |
| 67 | \( 1 + 7.69iT - 67T^{2} \) |
| 71 | \( 1 - 7.77iT - 71T^{2} \) |
| 73 | \( 1 - 4.96iT - 73T^{2} \) |
| 79 | \( 1 - 8.26T + 79T^{2} \) |
| 83 | \( 1 + 12.1iT - 83T^{2} \) |
| 89 | \( 1 + 3.17iT - 89T^{2} \) |
| 97 | \( 1 + 0.366iT - 97T^{2} \) |
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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
−11.51084073385104166140779900560, −10.75996221228885576310295057559, −9.334993133313735702456274576006, −8.858493622347315968041743396187, −7.57242326495139937424523199187, −6.90209009395338836922396850480, −5.65281949626883227011589714281, −5.03736984295245179091591982113, −2.84217085784167253917115284871, −2.33839181447394994229310234336,
1.08251451617496521826429578685, 2.88686809246881298936984185930, 3.67680226511296515685380892496, 5.25416188776818165346682147196, 6.44118203975022582539121551901, 7.35464784482550877880711985769, 8.129506315895779535448316035113, 9.374253691515365983075286851266, 10.48571725695364794067240817048, 10.91873591840496702579403358895