L(s) = 1 | + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.707 − 0.707i)7-s + (0.707 − 0.707i)8-s + i·9-s + 11-s + (−0.866 − 0.500i)14-s + (0.500 − 0.866i)16-s + (0.258 + 0.965i)18-s + (0.965 − 0.258i)22-s + (0.707 − 0.707i)23-s + (−0.965 − 0.258i)28-s − 1.73·29-s + (0.258 − 0.965i)32-s + (0.499 + 0.866i)36-s + (0.707 + 0.707i)37-s + ⋯ |
L(s) = 1 | + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.707 − 0.707i)7-s + (0.707 − 0.707i)8-s + i·9-s + 11-s + (−0.866 − 0.500i)14-s + (0.500 − 0.866i)16-s + (0.258 + 0.965i)18-s + (0.965 − 0.258i)22-s + (0.707 − 0.707i)23-s + (−0.965 − 0.258i)28-s − 1.73·29-s + (0.258 − 0.965i)32-s + (0.499 + 0.866i)36-s + (0.707 + 0.707i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.924455902\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.924455902\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 + 0.258i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
good | 3 | \( 1 - iT^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 29 | \( 1 + 1.73T + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + (1.41 - 1.41i)T - iT^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + (1.22 - 1.22i)T - iT^{2} \) |
| 71 | \( 1 - 1.73iT - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 - 1.73T + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 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
−9.885606556822739165542401933310, −9.036095820558415054602566116362, −7.78698496238020185561543825565, −7.03840942925624014084061344919, −6.37063204274593170195998760935, −5.41728060393882104276126082813, −4.46907491015074442792857825478, −3.73635276484342410361127850386, −2.73043743462125967300242324941, −1.47228903260477238735748361695,
1.78771350263553708087260607097, 3.21234037999579597618764266069, 3.65931690518395841056680973446, 4.83212742528104612060734788157, 5.86075846558359032359825282561, 6.38848536549007603383273534572, 7.09824385048030977965453513134, 8.107033659108864867954027910685, 9.309101440152034169972001935060, 9.463890025577830447473145459291