| L(s) = 1 | + 7-s + 4·13-s + 2·17-s − 2·19-s − 2·23-s + 4·29-s + 10·31-s − 6·37-s − 6·41-s + 4·43-s + 12·47-s + 49-s + 6·53-s + 12·59-s + 6·61-s − 4·67-s − 4·71-s + 8·73-s − 4·83-s + 18·89-s + 4·91-s + 12·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 1.10·13-s + 0.485·17-s − 0.458·19-s − 0.417·23-s + 0.742·29-s + 1.79·31-s − 0.986·37-s − 0.937·41-s + 0.609·43-s + 1.75·47-s + 1/7·49-s + 0.824·53-s + 1.56·59-s + 0.768·61-s − 0.488·67-s − 0.474·71-s + 0.936·73-s − 0.439·83-s + 1.90·89-s + 0.419·91-s + 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.100479725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.100479725\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−14.46195835273014, −13.92626533567492, −13.55435387414773, −13.13708867894422, −12.21234938917027, −12.08245761841708, −11.48916972512805, −10.85122301270658, −10.28163312697017, −10.11640296790344, −9.178874013191784, −8.667366981727084, −8.286322937905322, −7.791286996175430, −6.995732580985304, −6.551071587760869, −5.928438684968578, −5.398762379449983, −4.743064850909444, −4.044144195293780, −3.633020213980262, −2.759314034848356, −2.163397068779974, −1.264686868709002, −0.6846105573065361,
0.6846105573065361, 1.264686868709002, 2.163397068779974, 2.759314034848356, 3.633020213980262, 4.044144195293780, 4.743064850909444, 5.398762379449983, 5.928438684968578, 6.551071587760869, 6.995732580985304, 7.791286996175430, 8.286322937905322, 8.667366981727084, 9.178874013191784, 10.11640296790344, 10.28163312697017, 10.85122301270658, 11.48916972512805, 12.08245761841708, 12.21234938917027, 13.13708867894422, 13.55435387414773, 13.92626533567492, 14.46195835273014
