L(s) = 1 | + (−1.36 + 0.366i)3-s + (0.965 + 0.258i)5-s + (0.707 + 0.707i)7-s + (0.866 − 0.5i)9-s + (−0.866 − 0.5i)11-s + (0.707 + 0.707i)13-s − 1.41·15-s + (0.5 + 0.866i)19-s + (−1.22 − 0.707i)21-s + (0.965 + 0.258i)23-s + (0.866 + 0.499i)25-s − 1.41·29-s + (0.707 − 1.22i)31-s + (1.36 + 0.366i)33-s + (0.500 + 0.866i)35-s + ⋯ |
L(s) = 1 | + (−1.36 + 0.366i)3-s + (0.965 + 0.258i)5-s + (0.707 + 0.707i)7-s + (0.866 − 0.5i)9-s + (−0.866 − 0.5i)11-s + (0.707 + 0.707i)13-s − 1.41·15-s + (0.5 + 0.866i)19-s + (−1.22 − 0.707i)21-s + (0.965 + 0.258i)23-s + (0.866 + 0.499i)25-s − 1.41·29-s + (0.707 − 1.22i)31-s + (1.36 + 0.366i)33-s + (0.500 + 0.866i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.290 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.290 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9274375664\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9274375664\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.965 - 0.258i)T \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 3 | \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 17 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + 1.41T + T^{2} \) |
| 31 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + 1.41T + T^{2} \) |
| 73 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-1 - i)T + iT^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 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
−9.490320929101457442205979182651, −8.738604720512076837679738091727, −7.83047003478993126905654182991, −6.76108626129760079766243586617, −5.98535507863134921666557295256, −5.45264575536409168232010794156, −5.03913223893514254160410189461, −3.79664205411421427978286368843, −2.49952234326893147968702692859, −1.39879907749386944899217093563,
0.855205607572913191636577307476, 1.83462370145308869109352598509, 3.21392739673275333801016280113, 4.73923534229900782342140582023, 5.18036913223254484471754630209, 5.72792185655831557275928366841, 6.82080180452019531571283369166, 7.17343703533566427597422468183, 8.283499744621992372040428168159, 9.041424512864309204030323676729