| L(s) = 1 | + 2-s + 4-s + (2.23 − 1.41i)7-s + 8-s + 1.41i·11-s − 0.926·13-s + (2.23 − 1.41i)14-s + 16-s − 2.23i·17-s − 7.63i·19-s + 1.41i·22-s + 23-s − 0.926·26-s + (2.23 − 1.41i)28-s − 0.757i·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + (0.845 − 0.534i)7-s + 0.353·8-s + 0.426i·11-s − 0.256·13-s + (0.597 − 0.377i)14-s + 0.250·16-s − 0.542i·17-s − 1.75i·19-s + 0.301i·22-s + 0.208·23-s − 0.181·26-s + (0.422 − 0.267i)28-s − 0.140i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.192398685\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.192398685\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.23 + 1.41i)T \) |
| good | 11 | \( 1 - 1.41iT - 11T^{2} \) |
| 13 | \( 1 + 0.926T + 13T^{2} \) |
| 17 | \( 1 + 2.23iT - 17T^{2} \) |
| 19 | \( 1 + 7.63iT - 19T^{2} \) |
| 23 | \( 1 - T + 23T^{2} \) |
| 29 | \( 1 + 0.757iT - 29T^{2} \) |
| 31 | \( 1 + 4.08iT - 31T^{2} \) |
| 37 | \( 1 - 2.82iT - 37T^{2} \) |
| 41 | \( 1 - 8.56T + 41T^{2} \) |
| 43 | \( 1 + 3.58iT - 43T^{2} \) |
| 47 | \( 1 - 1.30iT - 47T^{2} \) |
| 53 | \( 1 - 8.07T + 53T^{2} \) |
| 59 | \( 1 + 7.25T + 59T^{2} \) |
| 61 | \( 1 + 0.926iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 15.6iT - 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 + 14.3iT - 83T^{2} \) |
| 89 | \( 1 - 2.61T + 89T^{2} \) |
| 97 | \( 1 - 0.542T + 97T^{2} \) |
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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
−8.543564384582761225793051565292, −7.44043419365128599530770341839, −7.27427504373926763259451677054, −6.29098405693889517259993328045, −5.30652162313399700602993636286, −4.67053212285144311455243651758, −4.11860257275618043111956043719, −2.90746544979422295273135663843, −2.12094465379866934711623394134, −0.811350011978743316061453134382,
1.34991033559655165765495886466, 2.24992848327993913588066033395, 3.30265025529125945214350900673, 4.13336154770751772102268930643, 4.96843829142805050237046161272, 5.75241252348345247993384919327, 6.21567423046819211608926096830, 7.35735014431849800587988288287, 7.972218724067222347796060461253, 8.632496792650311944668362068227
