L(s) = 1 | − 0.710i·2-s + 2.40·3-s + 1.49·4-s − 1.28i·5-s − 1.71i·6-s + i·7-s − 2.48i·8-s + 2.79·9-s − 0.916·10-s + 2.40i·11-s + 3.60·12-s + 0.710·14-s − 3.10i·15-s + 1.22·16-s − 3.90·17-s − 1.98i·18-s + ⋯ |
L(s) = 1 | − 0.502i·2-s + 1.39·3-s + 0.747·4-s − 0.576i·5-s − 0.698i·6-s + 0.377i·7-s − 0.877i·8-s + 0.932·9-s − 0.289·10-s + 0.726i·11-s + 1.03·12-s + 0.189·14-s − 0.801i·15-s + 0.306·16-s − 0.946·17-s − 0.468i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.203627540\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.203627540\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 - iT \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 0.710iT - 2T^{2} \) |
| 3 | \( 1 - 2.40T + 3T^{2} \) |
| 5 | \( 1 + 1.28iT - 5T^{2} \) |
| 11 | \( 1 - 2.40iT - 11T^{2} \) |
| 17 | \( 1 + 3.90T + 17T^{2} \) |
| 19 | \( 1 + 5.89iT - 19T^{2} \) |
| 23 | \( 1 - 6.32T + 23T^{2} \) |
| 29 | \( 1 - 5.61T + 29T^{2} \) |
| 31 | \( 1 - 2.20iT - 31T^{2} \) |
| 37 | \( 1 - 5.11iT - 37T^{2} \) |
| 41 | \( 1 + 7.78iT - 41T^{2} \) |
| 43 | \( 1 + 0.289T + 43T^{2} \) |
| 47 | \( 1 + 1.27iT - 47T^{2} \) |
| 53 | \( 1 + 13.6T + 53T^{2} \) |
| 59 | \( 1 - 4.03iT - 59T^{2} \) |
| 61 | \( 1 + 4.60T + 61T^{2} \) |
| 67 | \( 1 - 7.57iT - 67T^{2} \) |
| 71 | \( 1 - 7.22iT - 71T^{2} \) |
| 73 | \( 1 - 15.0iT - 73T^{2} \) |
| 79 | \( 1 + 9.30T + 79T^{2} \) |
| 83 | \( 1 + 1.36iT - 83T^{2} \) |
| 89 | \( 1 - 0.899iT - 89T^{2} \) |
| 97 | \( 1 - 15.6iT - 97T^{2} \) |
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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
−9.407387237018452574772495359400, −8.938905077362198583654801505225, −8.234879034799678421009227633621, −7.12603914947670637176300413813, −6.68092967243334126084322667488, −5.12347021427949356443999595522, −4.23372437368351428300852439049, −2.94126698529732879537382977210, −2.52610896471531388508723719255, −1.34833379215562490137303545141,
1.69447839823033993183584966495, 2.86690805806240668517075588308, 3.32410106011350187701140812195, 4.65464440274907922988346947922, 6.02846620498842586020145066271, 6.69262135121475911923314172343, 7.54776564646867351967360644132, 8.142417933836834574792765759496, 8.833487791035478031632696224334, 9.746096004740673668078135790748