L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 4·7-s + 8-s + 9-s + 12-s − 2.47·13-s + 4·14-s + 16-s + 2.47·17-s + 18-s + 2·19-s + 4·21-s − 23-s + 24-s − 2.47·26-s + 27-s + 4·28-s − 0.472·29-s + 32-s + 2.47·34-s + 36-s + 0.472·37-s + 2·38-s − 2.47·39-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s + 1.51·7-s + 0.353·8-s + 0.333·9-s + 0.288·12-s − 0.685·13-s + 1.06·14-s + 0.250·16-s + 0.599·17-s + 0.235·18-s + 0.458·19-s + 0.872·21-s − 0.208·23-s + 0.204·24-s − 0.484·26-s + 0.192·27-s + 0.755·28-s − 0.0876·29-s + 0.176·32-s + 0.423·34-s + 0.166·36-s + 0.0776·37-s + 0.324·38-s − 0.395·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.577084300\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.577084300\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 - 4T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 2.47T + 13T^{2} \) |
| 17 | \( 1 - 2.47T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 29 | \( 1 + 0.472T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 0.472T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 4.94T + 47T^{2} \) |
| 53 | \( 1 + 8.94T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 0.472T + 61T^{2} \) |
| 67 | \( 1 + 4.94T + 67T^{2} \) |
| 71 | \( 1 + 7.52T + 71T^{2} \) |
| 73 | \( 1 - 4.94T + 73T^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 - 1.52T + 83T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 + 13.4T + 97T^{2} \) |
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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
−8.424193300893048907259665118448, −7.59777646235107738865208239945, −7.47318900460785230539590292658, −6.22210519152707673983040126127, −5.37714618032900714608010628465, −4.71980287745109039196904565568, −4.05934126312131973655591662643, −3.01309112498491954557173784436, −2.16627085825204070327855970572, −1.24362686956506664914624916094,
1.24362686956506664914624916094, 2.16627085825204070327855970572, 3.01309112498491954557173784436, 4.05934126312131973655591662643, 4.71980287745109039196904565568, 5.37714618032900714608010628465, 6.22210519152707673983040126127, 7.47318900460785230539590292658, 7.59777646235107738865208239945, 8.424193300893048907259665118448