L(s) = 1 | + (−0.541 − 0.541i)3-s + (0.382 + 0.923i)5-s + (−0.707 + 0.707i)7-s − 0.414i·9-s + (−1.30 + 1.30i)13-s + (0.292 − 0.707i)15-s − 0.765·19-s + 0.765·21-s + (−1 − i)23-s + (−0.707 + 0.707i)25-s + (−0.765 + 0.765i)27-s + (−0.923 − 0.382i)35-s + 1.41·39-s + (0.382 − 0.158i)45-s − 1.00i·49-s + ⋯ |
L(s) = 1 | + (−0.541 − 0.541i)3-s + (0.382 + 0.923i)5-s + (−0.707 + 0.707i)7-s − 0.414i·9-s + (−1.30 + 1.30i)13-s + (0.292 − 0.707i)15-s − 0.765·19-s + 0.765·21-s + (−1 − i)23-s + (−0.707 + 0.707i)25-s + (−0.765 + 0.765i)27-s + (−0.923 − 0.382i)35-s + 1.41·39-s + (0.382 − 0.158i)45-s − 1.00i·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3400600083\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3400600083\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.382 - 0.923i)T \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
good | 3 | \( 1 + (0.541 + 0.541i)T + iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (1.30 - 1.30i)T - iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + 0.765T + T^{2} \) |
| 23 | \( 1 + (1 + i)T + iT^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + 1.84T + T^{2} \) |
| 61 | \( 1 + 1.84T + T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 - 1.41iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 - 1.41T + T^{2} \) |
| 83 | \( 1 + (-1.30 - 1.30i)T + iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−9.489663712007692713915980104483, −9.000769393420383263161743730243, −7.77613250295489786391072147495, −6.93583882189850707429775629027, −6.39647151874582020736247126310, −6.00949974675568307281221977881, −4.84325464609802826228034615150, −3.76946117308220339066733935572, −2.62739367168260214407007785523, −1.94646993947349081074951999176,
0.22796811593665227942670078064, 1.90505779635810470906327441386, 3.16212793659342455736757292410, 4.29509538243420182448564284186, 4.89638601533322967133969923012, 5.66546794319387379396375129776, 6.33110778267927700646809907521, 7.66730714646442635684956947583, 7.890833121614868531944904907455, 9.204117750912013357430435574572