L(s) = 1 | + (1.40 − 0.114i)2-s + (1.97 − 0.321i)4-s + 1.12i·5-s + 7-s + (2.74 − 0.678i)8-s + (0.128 + 1.59i)10-s + 4.76i·11-s + 0.456i·13-s + (1.40 − 0.114i)14-s + (3.79 − 1.26i)16-s − 0.415·17-s − 7.63i·19-s + (0.362 + 2.22i)20-s + (0.543 + 6.71i)22-s − 1.58·23-s + ⋯ |
L(s) = 1 | + (0.996 − 0.0806i)2-s + (0.986 − 0.160i)4-s + 0.504i·5-s + 0.377·7-s + (0.970 − 0.239i)8-s + (0.0407 + 0.503i)10-s + 1.43i·11-s + 0.126i·13-s + (0.376 − 0.0304i)14-s + (0.948 − 0.317i)16-s − 0.100·17-s − 1.75i·19-s + (0.0811 + 0.498i)20-s + (0.115 + 1.43i)22-s − 0.330·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.72720 + 0.331923i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.72720 + 0.331923i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.40 + 0.114i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 - 1.12iT - 5T^{2} \) |
| 11 | \( 1 - 4.76iT - 11T^{2} \) |
| 13 | \( 1 - 0.456iT - 13T^{2} \) |
| 17 | \( 1 + 0.415T + 17T^{2} \) |
| 19 | \( 1 + 7.63iT - 19T^{2} \) |
| 23 | \( 1 + 1.58T + 23T^{2} \) |
| 29 | \( 1 + 6.72iT - 29T^{2} \) |
| 31 | \( 1 + 5.89T + 31T^{2} \) |
| 37 | \( 1 - 5.89iT - 37T^{2} \) |
| 41 | \( 1 - 0.415T + 41T^{2} \) |
| 43 | \( 1 - 9.43iT - 43T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 + 7.63iT - 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 + 1.80iT - 61T^{2} \) |
| 67 | \( 1 + 8.09iT - 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 + 3.34T + 73T^{2} \) |
| 79 | \( 1 + 4.83T + 79T^{2} \) |
| 83 | \( 1 + 5.53iT - 83T^{2} \) |
| 89 | \( 1 + 4.92T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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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
−11.23171246911377278564778226587, −10.26018684829271363477559873193, −9.403554078571833659579657016436, −7.943384430751275810298933919535, −7.04971036118181880435626049908, −6.41570332406215325822000921135, −5.00570832815294868399610530159, −4.42073875787453191274738032493, −3.00232858290797988011737880139, −1.92146194737866490544514208631,
1.54828611847165281916656256033, 3.17910039594386657395572919818, 4.10103043067736742002546404459, 5.37949010843785337960412094676, 5.87674141930311228373888966753, 7.12999354779534989655671629363, 8.144189992303604575116209533101, 8.869376384141027822873997772291, 10.37137589871760183763488548060, 10.99627749051595186608476081206