| L(s) = 1 | + 1.23·3-s + 5.23·7-s − 1.47·9-s + 6.47·21-s + 7.70·23-s − 5.52·27-s + 6·29-s + 4.47·41-s − 6.76·43-s + 0.291·47-s + 20.4·49-s − 13.4·61-s − 7.70·63-s − 14.1·67-s + 9.52·69-s − 2.41·81-s + 4.29·83-s + 7.41·87-s + 6·89-s + 18·101-s − 2.18·103-s + 19.7·107-s + 13.4·109-s + ⋯ |
| L(s) = 1 | + 0.713·3-s + 1.97·7-s − 0.490·9-s + 1.41·21-s + 1.60·23-s − 1.06·27-s + 1.11·29-s + 0.698·41-s − 1.03·43-s + 0.0425·47-s + 2.91·49-s − 1.71·61-s − 0.971·63-s − 1.73·67-s + 1.14·69-s − 0.268·81-s + 0.471·83-s + 0.795·87-s + 0.635·89-s + 1.79·101-s − 0.214·103-s + 1.90·107-s + 1.28·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.681915527\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.681915527\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 1.23T + 3T^{2} \) |
| 7 | \( 1 - 5.23T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 7.70T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 4.47T + 41T^{2} \) |
| 43 | \( 1 + 6.76T + 43T^{2} \) |
| 47 | \( 1 - 0.291T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 4.29T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−8.981021072051790332711909896389, −8.715682100582172526057577956387, −7.86922279576880230218534033104, −7.35729692236798593523829620330, −6.10430167146131886553183041943, −5.06187783798438436590623467379, −4.55449588525619354671206126566, −3.27386988931851935633473820789, −2.31422058439203483863311661110, −1.24744484253273168861608210071,
1.24744484253273168861608210071, 2.31422058439203483863311661110, 3.27386988931851935633473820789, 4.55449588525619354671206126566, 5.06187783798438436590623467379, 6.10430167146131886553183041943, 7.35729692236798593523829620330, 7.86922279576880230218534033104, 8.715682100582172526057577956387, 8.981021072051790332711909896389
