L(s) = 1 | + (1.03 + 1.03i)3-s + 1.49·7-s − 0.836i·9-s + (−0.423 − 0.423i)11-s + (−1.85 − 1.85i)13-s − 6.50i·17-s + (−1.75 + 1.75i)19-s + (1.55 + 1.55i)21-s + 7.19·23-s + (3.99 − 3.99i)27-s + (6.57 − 6.57i)29-s + 6.75·31-s − 0.880i·33-s + (1.95 − 1.95i)37-s − 3.86i·39-s + ⋯ |
L(s) = 1 | + (0.600 + 0.600i)3-s + 0.565·7-s − 0.278i·9-s + (−0.127 − 0.127i)11-s + (−0.515 − 0.515i)13-s − 1.57i·17-s + (−0.403 + 0.403i)19-s + (0.339 + 0.339i)21-s + 1.49·23-s + (0.767 − 0.767i)27-s + (1.22 − 1.22i)29-s + 1.21·31-s − 0.153i·33-s + (0.321 − 0.321i)37-s − 0.618i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.295i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 + 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.190515315\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.190515315\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-1.03 - 1.03i)T + 3iT^{2} \) |
| 7 | \( 1 - 1.49T + 7T^{2} \) |
| 11 | \( 1 + (0.423 + 0.423i)T + 11iT^{2} \) |
| 13 | \( 1 + (1.85 + 1.85i)T + 13iT^{2} \) |
| 17 | \( 1 + 6.50iT - 17T^{2} \) |
| 19 | \( 1 + (1.75 - 1.75i)T - 19iT^{2} \) |
| 23 | \( 1 - 7.19T + 23T^{2} \) |
| 29 | \( 1 + (-6.57 + 6.57i)T - 29iT^{2} \) |
| 31 | \( 1 - 6.75T + 31T^{2} \) |
| 37 | \( 1 + (-1.95 + 1.95i)T - 37iT^{2} \) |
| 41 | \( 1 - 7.70iT - 41T^{2} \) |
| 43 | \( 1 + (6.13 - 6.13i)T - 43iT^{2} \) |
| 47 | \( 1 + 6.65iT - 47T^{2} \) |
| 53 | \( 1 + (5.29 - 5.29i)T - 53iT^{2} \) |
| 59 | \( 1 + (-5.91 - 5.91i)T + 59iT^{2} \) |
| 61 | \( 1 + (1.43 - 1.43i)T - 61iT^{2} \) |
| 67 | \( 1 + (6.35 + 6.35i)T + 67iT^{2} \) |
| 71 | \( 1 - 4.08iT - 71T^{2} \) |
| 73 | \( 1 - 2.43T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 + (2.81 + 2.81i)T + 83iT^{2} \) |
| 89 | \( 1 - 10.5iT - 89T^{2} \) |
| 97 | \( 1 + 18.1iT - 97T^{2} \) |
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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
−9.538897086475054827988984462090, −8.518133476711237954322466973590, −8.017618373170479816056630422634, −7.02958740336680285007800722473, −6.14905970708917358706662790889, −4.89235412618044179238530020080, −4.53501329228287139257704526972, −3.16543463493575522250323692672, −2.62242668607749000224286073951, −0.874795083498088180037686368775,
1.36540931063682102989564774489, 2.26048693967147118290024005660, 3.25943688161442284320149495339, 4.56803959624276012069036433555, 5.14858675889988411688362472095, 6.51492818553188645270655086291, 7.02755604256058591510506830485, 8.047226704274358551997704784564, 8.450585677546525609082574273572, 9.214419679408182623498678336968