L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + (−0.707 + 0.707i)5-s + 1.41·7-s + (−0.707 − 0.707i)8-s − i·9-s + 1.00i·10-s + (0.707 + 0.707i)13-s + (1.00 − 1.00i)14-s − 1.00·16-s + (−0.707 − 0.707i)18-s + (0.707 + 0.707i)20-s − 1.00i·25-s + 1.00·26-s − 1.41i·28-s + (−1 − i)29-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + (−0.707 + 0.707i)5-s + 1.41·7-s + (−0.707 − 0.707i)8-s − i·9-s + 1.00i·10-s + (0.707 + 0.707i)13-s + (1.00 − 1.00i)14-s − 1.00·16-s + (−0.707 − 0.707i)18-s + (0.707 + 0.707i)20-s − 1.00i·25-s + 1.00·26-s − 1.41i·28-s + (−1 − i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.478810281\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.478810281\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
| 13 | \( 1 + (-0.707 - 0.707i)T \) |
good | 3 | \( 1 + iT^{2} \) |
| 7 | \( 1 - 1.41T + T^{2} \) |
| 11 | \( 1 - iT^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (1 + i)T + iT^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 - 1.41iT - T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + (-1 - i)T + iT^{2} \) |
| 67 | \( 1 + (1.41 - 1.41i)T - iT^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.41iT - T^{2} \) |
| 79 | \( 1 - 2iT - T^{2} \) |
| 83 | \( 1 + (1.41 - 1.41i)T - iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + 1.41T + T^{2} \) |
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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
−10.22599389611638223906200185872, −9.257212732592679157935367351049, −8.383800961912270556035506241003, −7.36300456147231296634852935355, −6.45947604000961988501938732943, −5.59367200220802434548998603140, −4.32002358656696978796945152672, −3.91118033309518516630462113715, −2.68980435341209283407962396602, −1.39650715523926000449103022997,
1.80398406833872364908613193021, 3.38727970396561220770471484071, 4.41020746037465488779234338140, 5.09998631041694403050545089909, 5.66881651040936602670631241031, 7.13934297188828263054490812346, 7.83719368044416238617072164309, 8.322435289836134849541167417871, 8.971152740236731555063579123132, 10.57528943162120993551082102831