L(s) = 1 | − 2-s + 4-s − 2·5-s − 4·7-s − 8-s + 2·10-s + 11-s − 6·13-s + 4·14-s + 16-s − 2·17-s + 4·19-s − 2·20-s − 22-s − 4·23-s − 25-s + 6·26-s − 4·28-s − 6·29-s − 32-s + 2·34-s + 8·35-s + 6·37-s − 4·38-s + 2·40-s + 6·41-s + 4·43-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 1.51·7-s − 0.353·8-s + 0.632·10-s + 0.301·11-s − 1.66·13-s + 1.06·14-s + 1/4·16-s − 0.485·17-s + 0.917·19-s − 0.447·20-s − 0.213·22-s − 0.834·23-s − 1/5·25-s + 1.17·26-s − 0.755·28-s − 1.11·29-s − 0.176·32-s + 0.342·34-s + 1.35·35-s + 0.986·37-s − 0.648·38-s + 0.316·40-s + 0.937·41-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 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 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−12.06831548180719701984563530619, −10.95942742879596868640115475077, −9.695426551233672906693226319091, −9.331429543089205578507139477048, −7.76149770434504125566929280206, −7.14262372141687320205238561411, −5.90428105253920776614679871565, −4.09304077381815838984743047096, −2.74090516225731871253746662847, 0,
2.74090516225731871253746662847, 4.09304077381815838984743047096, 5.90428105253920776614679871565, 7.14262372141687320205238561411, 7.76149770434504125566929280206, 9.331429543089205578507139477048, 9.695426551233672906693226319091, 10.95942742879596868640115475077, 12.06831548180719701984563530619