L(s) = 1 | + 2.56·3-s + (−2 − i)5-s + (−2 − 1.73i)7-s + 3.56·9-s − 0.972·11-s + 0.561i·13-s + (−5.12 − 2.56i)15-s − 4.43·17-s + 1.12i·19-s + (−5.12 − 4.43i)21-s + 1.12·23-s + (3 + 4i)25-s + 1.43·27-s − 4.43i·29-s − 8.87·31-s + ⋯ |
L(s) = 1 | + 1.47·3-s + (−0.894 − 0.447i)5-s + (−0.755 − 0.654i)7-s + 1.18·9-s − 0.293·11-s + 0.155i·13-s + (−1.32 − 0.661i)15-s − 1.07·17-s + 0.257i·19-s + (−1.11 − 0.968i)21-s + 0.234·23-s + (0.600 + 0.800i)25-s + 0.276·27-s − 0.823i·29-s − 1.59·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.131i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5912809098\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5912809098\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (2 + i)T \) |
| 7 | \( 1 + (2 + 1.73i)T \) |
good | 3 | \( 1 - 2.56T + 3T^{2} \) |
| 11 | \( 1 + 0.972T + 11T^{2} \) |
| 13 | \( 1 - 0.561iT - 13T^{2} \) |
| 17 | \( 1 + 4.43T + 17T^{2} \) |
| 19 | \( 1 - 1.12iT - 19T^{2} \) |
| 23 | \( 1 - 1.12T + 23T^{2} \) |
| 29 | \( 1 + 4.43iT - 29T^{2} \) |
| 31 | \( 1 + 8.87T + 31T^{2} \) |
| 37 | \( 1 + 8.87T + 37T^{2} \) |
| 41 | \( 1 + 8.87iT - 41T^{2} \) |
| 43 | \( 1 - 1.94iT - 43T^{2} \) |
| 47 | \( 1 - 0.972iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 - 1.12T + 61T^{2} \) |
| 67 | \( 1 + 6.92iT - 67T^{2} \) |
| 71 | \( 1 - 14.2iT - 71T^{2} \) |
| 73 | \( 1 + 6.92T + 73T^{2} \) |
| 79 | \( 1 + 7.68iT - 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 8.87iT - 89T^{2} \) |
| 97 | \( 1 + 9.41T + 97T^{2} \) |
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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
−8.838683824502558442651965195673, −7.898547520346032115288444538593, −7.35187340195017965982348612220, −6.67761397969854464119827856984, −5.34569932303286531516832774331, −4.17544854390150378671152919915, −3.75969191151589261910754122305, −2.91117406896261707510841230486, −1.81335758796308228276985472492, −0.15184062216021786378532492376,
1.97963765698320278495639753239, 2.91755729050251133217303706910, 3.38617844387408714681503668866, 4.28143402940982650295159523691, 5.39848228025544793574149800371, 6.62514572753799185838999555361, 7.19491725035965252592349848434, 7.956622959994932955103413085188, 8.860963157346789632396267999321, 8.954680925578622371784417158376