| L(s) = 1 | + i·2-s + (−0.366 + 0.366i)3-s − 4-s + (−0.366 − 0.366i)6-s − i·8-s + 0.732i·9-s + (0.366 + 0.366i)11-s + (0.366 − 0.366i)12-s + 16-s + (0.866 − 0.5i)17-s − 0.732·18-s + i·19-s + (−0.366 + 0.366i)22-s + (0.366 + 0.366i)24-s + (−0.633 − 0.633i)27-s + ⋯ |
| L(s) = 1 | + i·2-s + (−0.366 + 0.366i)3-s − 4-s + (−0.366 − 0.366i)6-s − i·8-s + 0.732i·9-s + (0.366 + 0.366i)11-s + (0.366 − 0.366i)12-s + 16-s + (0.866 − 0.5i)17-s − 0.732·18-s + i·19-s + (−0.366 + 0.366i)22-s + (0.366 + 0.366i)24-s + (−0.633 − 0.633i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9220330580\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9220330580\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 17 | \( 1 + (-0.866 + 0.5i)T \) |
| good | 3 | \( 1 + (0.366 - 0.366i)T - iT^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + (-0.366 - 0.366i)T + iT^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 19 | \( 1 - iT - T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 + iT^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (-1.36 - 1.36i)T + iT^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - iT^{2} \) |
| 67 | \( 1 + 1.73T + T^{2} \) |
| 71 | \( 1 + iT^{2} \) |
| 73 | \( 1 + (0.366 - 0.366i)T - iT^{2} \) |
| 79 | \( 1 - iT^{2} \) |
| 83 | \( 1 - 1.73iT - T^{2} \) |
| 89 | \( 1 + 1.73T + T^{2} \) |
| 97 | \( 1 + (-1 + i)T - iT^{2} \) |
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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
−9.112022910835953426892962224266, −8.057600306041070978249536376799, −7.73529026953370710779386083829, −6.89816602989321957195655656041, −5.96925073901235451876086820366, −5.50083933275351093701190138639, −4.61669434896429059116152178468, −4.05730827886576024838729012760, −2.90524645947023410677566712235, −1.36289177815156283660929213053,
0.65242536691495196331825247305, 1.66636429725863629061618210899, 2.85518287969901363076571836770, 3.63868874067601640540385027634, 4.40484200849667755212365473994, 5.46723205670754353232063421029, 6.02518321651618348709777286731, 6.99022823751886683244303780653, 7.79021106604353350556774956084, 8.776999280772544786056639946130
