L(s) = 1 | + (0.707 + 0.707i)3-s + (0.707 + 0.707i)5-s + 11-s + (−0.707 − 0.707i)13-s + 1.00i·15-s + (−0.707 + 0.707i)17-s + 1.41i·19-s + 1.00i·25-s + (0.707 − 0.707i)27-s − i·29-s + 1.41·31-s + (0.707 + 0.707i)33-s − 1.00i·39-s − 1.41·41-s + (−1 − i)43-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)3-s + (0.707 + 0.707i)5-s + 11-s + (−0.707 − 0.707i)13-s + 1.00i·15-s + (−0.707 + 0.707i)17-s + 1.41i·19-s + 1.00i·25-s + (0.707 − 0.707i)27-s − i·29-s + 1.41·31-s + (0.707 + 0.707i)33-s − 1.00i·39-s − 1.41·41-s + (−1 − i)43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.661219452\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.661219452\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.707 - 0.707i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 17 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 19 | \( 1 - 1.41iT - T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 + iT - T^{2} \) |
| 31 | \( 1 - 1.41T + T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + 1.41T + T^{2} \) |
| 43 | \( 1 + (1 + i)T + iT^{2} \) |
| 47 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 53 | \( 1 + (1 + i)T + iT^{2} \) |
| 59 | \( 1 - 1.41iT - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + (1 - i)T - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 - iT - T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
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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
−9.808676771574831404325543310557, −8.700757147792071855470255198362, −8.213996034213438079098300838178, −7.03852379862538316210765662644, −6.34978255089868026487900151360, −5.59075646136150993324910360243, −4.37322042557541065139711921262, −3.63696915767250977084305921486, −2.80856031165082129564887514783, −1.74746769945061205058032668569,
1.30331451197242421052466012012, 2.20330101542419209099883202251, 3.06473609289237927131530240221, 4.67065651444874476297985607706, 4.88833739359896774435953959533, 6.43128032781403388384822115981, 6.78455456199932045878558263786, 7.70939243015627270064818668767, 8.642214106721314138006257428671, 9.130574713424885829263015637640