| L(s) = 1 | + 1.41·3-s − 2.82·5-s − 4·7-s − 0.999·9-s + 1.41·11-s + 2.82·13-s − 4.00·15-s − 4·17-s − 7.07·19-s − 5.65·21-s − 4·23-s + 3.00·25-s − 5.65·27-s + 8.48·29-s − 8·31-s + 2.00·33-s + 11.3·35-s + 2.82·37-s + 4.00·39-s + 2·41-s + 4.24·43-s + 2.82·45-s + 9·49-s − 5.65·51-s + 2.82·53-s − 4.00·55-s − 10.0·57-s + ⋯ |
| L(s) = 1 | + 0.816·3-s − 1.26·5-s − 1.51·7-s − 0.333·9-s + 0.426·11-s + 0.784·13-s − 1.03·15-s − 0.970·17-s − 1.62·19-s − 1.23·21-s − 0.834·23-s + 0.600·25-s − 1.08·27-s + 1.57·29-s − 1.43·31-s + 0.348·33-s + 1.91·35-s + 0.464·37-s + 0.640·39-s + 0.312·41-s + 0.646·43-s + 0.421·45-s + 1.28·49-s − 0.792·51-s + 0.388·53-s − 0.539·55-s − 1.32·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 5 | \( 1 + 2.82T + 5T^{2} \) |
| 7 | \( 1 + 4T + 7T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + 7.07T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 8.48T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 2.82T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4.24T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 2.82T + 53T^{2} \) |
| 59 | \( 1 - 4.24T + 59T^{2} \) |
| 61 | \( 1 + 8.48T + 61T^{2} \) |
| 67 | \( 1 - 4.24T + 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 9.89T + 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 + 4T + 97T^{2} \) |
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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
−10.49574187132408215607831995199, −9.281525011802992645370023439182, −8.686721910286336691620343293808, −7.945112543853197062870817498787, −6.77474245661909789851894232306, −6.08610173063719762075708845260, −4.16725586767261259610490344282, −3.64602790789663783204629244220, −2.51934096660172210523376964460, 0,
2.51934096660172210523376964460, 3.64602790789663783204629244220, 4.16725586767261259610490344282, 6.08610173063719762075708845260, 6.77474245661909789851894232306, 7.945112543853197062870817498787, 8.686721910286336691620343293808, 9.281525011802992645370023439182, 10.49574187132408215607831995199
