| L(s) = 1 | − 3·11-s − 2·13-s − 4·17-s − 3·23-s − 29-s + 2·31-s + 7·37-s − 2·41-s + 43-s − 12·47-s − 6·53-s + 6·59-s + 6·61-s + 7·67-s − 3·71-s − 2·73-s + 5·79-s − 6·83-s + 18·89-s − 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.904·11-s − 0.554·13-s − 0.970·17-s − 0.625·23-s − 0.185·29-s + 0.359·31-s + 1.15·37-s − 0.312·41-s + 0.152·43-s − 1.75·47-s − 0.824·53-s + 0.781·59-s + 0.768·61-s + 0.855·67-s − 0.356·71-s − 0.234·73-s + 0.562·79-s − 0.658·83-s + 1.90·89-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.36161102597966, −13.02290273233410, −12.52755708443813, −12.01188948098994, −11.47649098666037, −11.09850278996693, −10.59271639615194, −10.08976296078439, −9.581423465131629, −9.313289073774669, −8.485402517465953, −8.084308020955900, −7.836816737764557, −7.009327811852407, −6.731219118113558, −6.112965061860721, −5.532938677348117, −5.041468274487592, −4.515884750096555, −4.059536363416670, −3.294101398586763, −2.732742783022115, −2.210892216797193, −1.660215857866099, −0.6684622900443648, 0,
0.6684622900443648, 1.660215857866099, 2.210892216797193, 2.732742783022115, 3.294101398586763, 4.059536363416670, 4.515884750096555, 5.041468274487592, 5.532938677348117, 6.112965061860721, 6.731219118113558, 7.009327811852407, 7.836816737764557, 8.084308020955900, 8.485402517465953, 9.313289073774669, 9.581423465131629, 10.08976296078439, 10.59271639615194, 11.09850278996693, 11.47649098666037, 12.01188948098994, 12.52755708443813, 13.02290273233410, 13.36161102597966
