L(s) = 1 | − 1.41·3-s − 7-s − 0.999·9-s + 1.41·11-s + 1.41·19-s + 1.41·21-s + 4·23-s − 5·25-s + 5.65·27-s − 2.82·29-s + 4·31-s − 2.00·33-s − 2.82·37-s − 6·41-s + 4.24·43-s − 4·47-s + 49-s + 8.48·53-s − 2.00·57-s + 7.07·59-s + 5.65·61-s + 0.999·63-s − 1.41·67-s − 5.65·69-s − 8·71-s − 8·73-s + 7.07·75-s + ⋯ |
L(s) = 1 | − 0.816·3-s − 0.377·7-s − 0.333·9-s + 0.426·11-s + 0.324·19-s + 0.308·21-s + 0.834·23-s − 25-s + 1.08·27-s − 0.525·29-s + 0.718·31-s − 0.348·33-s − 0.464·37-s − 0.937·41-s + 0.646·43-s − 0.583·47-s + 0.142·49-s + 1.16·53-s − 0.264·57-s + 0.920·59-s + 0.724·61-s + 0.125·63-s − 0.172·67-s − 0.681·69-s − 0.949·71-s − 0.936·73-s + 0.816·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 1.41T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 2.82T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 2.82T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4.24T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 - 8.48T + 53T^{2} \) |
| 59 | \( 1 - 7.07T + 59T^{2} \) |
| 61 | \( 1 - 5.65T + 61T^{2} \) |
| 67 | \( 1 + 1.41T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 8T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 7.07T + 83T^{2} \) |
| 89 | \( 1 + 8T + 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−8.274713950974097415960695304009, −7.22516206329449014700240201827, −6.68742467776977706985105302065, −5.83259128796665629065391335799, −5.36637039550000827660304201777, −4.40463108894275388966515453718, −3.49910126923243332093908676315, −2.56702188832366835142854182208, −1.24145913859199717675211189361, 0,
1.24145913859199717675211189361, 2.56702188832366835142854182208, 3.49910126923243332093908676315, 4.40463108894275388966515453718, 5.36637039550000827660304201777, 5.83259128796665629065391335799, 6.68742467776977706985105302065, 7.22516206329449014700240201827, 8.274713950974097415960695304009