| L(s) = 1 | − 3-s + 2.89i·5-s + i·7-s + 9-s + 4.62i·11-s + (−2.28 − 2.78i)13-s − 2.89i·15-s + 2.68·17-s + 6.97i·19-s − i·21-s − 5.74·23-s − 3.40·25-s − 27-s − 9.63·29-s + 4.95i·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.29i·5-s + 0.377i·7-s + 0.333·9-s + 1.39i·11-s + (−0.633 − 0.773i)13-s − 0.748i·15-s + 0.650·17-s + 1.59i·19-s − 0.218i·21-s − 1.19·23-s − 0.681·25-s − 0.192·27-s − 1.78·29-s + 0.889i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.773 + 0.633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.773 + 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5757683299\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5757683299\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - iT \) |
| 13 | \( 1 + (2.28 + 2.78i)T \) |
| good | 5 | \( 1 - 2.89iT - 5T^{2} \) |
| 11 | \( 1 - 4.62iT - 11T^{2} \) |
| 17 | \( 1 - 2.68T + 17T^{2} \) |
| 19 | \( 1 - 6.97iT - 19T^{2} \) |
| 23 | \( 1 + 5.74T + 23T^{2} \) |
| 29 | \( 1 + 9.63T + 29T^{2} \) |
| 31 | \( 1 - 4.95iT - 31T^{2} \) |
| 37 | \( 1 - 3.11iT - 37T^{2} \) |
| 41 | \( 1 + 9.43iT - 41T^{2} \) |
| 43 | \( 1 - 2.54T + 43T^{2} \) |
| 47 | \( 1 + 9.87iT - 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 - 13.4iT - 59T^{2} \) |
| 61 | \( 1 + 6.69T + 61T^{2} \) |
| 67 | \( 1 - 5.56iT - 67T^{2} \) |
| 71 | \( 1 - 5.50iT - 71T^{2} \) |
| 73 | \( 1 - 6.20iT - 73T^{2} \) |
| 79 | \( 1 + 2.65T + 79T^{2} \) |
| 83 | \( 1 + 15.9iT - 83T^{2} \) |
| 89 | \( 1 - 2.24iT - 89T^{2} \) |
| 97 | \( 1 - 14.5iT - 97T^{2} \) |
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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
−8.810422184558529248962267453020, −7.70548601495478527832384808875, −7.38441800987397024210123012421, −6.72734357666006079098024357099, −5.63384471762245631424292615364, −5.53077875926115293297991813683, −4.19361445216302540853283078526, −3.52209548474789874776479528083, −2.45699294592085218125014770413, −1.70774312927512662094669414645,
0.19685385697536426192569054037, 1.00126814838472669839988258618, 2.15348446846338414219685144633, 3.45353697705754771497974578656, 4.34187155000481307290503286658, 4.90223935610123510228975769027, 5.72248721407307054948868349882, 6.24711557956504906322804077061, 7.30391579425554572042493334330, 7.899309031656078685161432627826
