| L(s) = 1 | + (1 + 2i)5-s + i·7-s + 3·9-s − 4i·13-s + 4i·17-s − 4·19-s − 8i·23-s + (−3 + 4i)25-s − 2·29-s − 8·31-s + (−2 + i)35-s − 8i·37-s + 6·41-s − 8i·43-s + (3 + 6i)45-s + ⋯ |
| L(s) = 1 | + (0.447 + 0.894i)5-s + 0.377i·7-s + 9-s − 1.10i·13-s + 0.970i·17-s − 0.917·19-s − 1.66i·23-s + (−0.600 + 0.800i)25-s − 0.371·29-s − 1.43·31-s + (−0.338 + 0.169i)35-s − 1.31i·37-s + 0.937·41-s − 1.21i·43-s + (0.447 + 0.894i)45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.15984 + 0.273802i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15984 + 0.273802i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1 - 2i)T \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 12iT - 97T^{2} \) |
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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
−12.96669995256079151075404019677, −12.56486098484899281556676128666, −10.79324829091812215006432572837, −10.45585803066577920268658596337, −9.172468262081889166146317305118, −7.85347736771192485577263380643, −6.70175068557014595121269523456, −5.65226771146563682921522801148, −3.93490860985937793236879916016, −2.27423480707383389006180586212,
1.67830909819770971279465816738, 4.03744434739196994906888648472, 5.12471343564457321137671406627, 6.62707383899723269380668387610, 7.73105409135580991251498714852, 9.187506760102842660251056287206, 9.732895421012609469365159122906, 11.10031189249695439529331808755, 12.18572119518809602416269541520, 13.19936138384739914382352658115
