L(s) = 1 | + (1.35 + 0.405i)2-s + (1.67 + 1.09i)4-s − 3.31i·5-s + i·7-s + (1.81 + 2.16i)8-s + (1.34 − 4.48i)10-s + 4.72·11-s − 4.97·13-s + (−0.405 + 1.35i)14-s + (1.58 + 3.67i)16-s − 0.484i·17-s + 2.29i·19-s + (3.63 − 5.53i)20-s + (6.40 + 1.91i)22-s − 7.97·23-s + ⋯ |
L(s) = 1 | + (0.958 + 0.286i)2-s + (0.835 + 0.549i)4-s − 1.48i·5-s + 0.377i·7-s + (0.643 + 0.765i)8-s + (0.424 − 1.41i)10-s + 1.42·11-s − 1.38·13-s + (−0.108 + 0.362i)14-s + (0.396 + 0.917i)16-s − 0.117i·17-s + 0.525i·19-s + (0.813 − 1.23i)20-s + (1.36 + 0.408i)22-s − 1.66·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0341i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0341i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.16464 + 0.0369601i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.16464 + 0.0369601i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.35 - 0.405i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 + 3.31iT - 5T^{2} \) |
| 11 | \( 1 - 4.72T + 11T^{2} \) |
| 13 | \( 1 + 4.97T + 13T^{2} \) |
| 17 | \( 1 + 0.484iT - 17T^{2} \) |
| 19 | \( 1 - 2.29iT - 19T^{2} \) |
| 23 | \( 1 + 7.97T + 23T^{2} \) |
| 29 | \( 1 + 1.41iT - 29T^{2} \) |
| 31 | \( 1 - 7.66iT - 31T^{2} \) |
| 37 | \( 1 + 2.39T + 37T^{2} \) |
| 41 | \( 1 + 6.55iT - 41T^{2} \) |
| 43 | \( 1 + 5.37iT - 43T^{2} \) |
| 47 | \( 1 + 6.21T + 47T^{2} \) |
| 53 | \( 1 - 1.00iT - 53T^{2} \) |
| 59 | \( 1 + 1.38T + 59T^{2} \) |
| 61 | \( 1 - 13.6T + 61T^{2} \) |
| 67 | \( 1 + 3.27iT - 67T^{2} \) |
| 71 | \( 1 + 3.34T + 71T^{2} \) |
| 73 | \( 1 + 2.10T + 73T^{2} \) |
| 79 | \( 1 - 12.0iT - 79T^{2} \) |
| 83 | \( 1 - 3.24T + 83T^{2} \) |
| 89 | \( 1 + 5.72iT - 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−12.12107275433841119703588902733, −11.79834332978403068366907688114, −10.07881733907071220330901049106, −8.996027296171327850303157369217, −8.112737833370005258629647941877, −6.89094191513584650601634236675, −5.69260840525412724197474595860, −4.80689709861664701538979324409, −3.82376832259150188931083497172, −1.89090127530682706920662359393,
2.20117831659463861496955943204, 3.44830454432889321846769754319, 4.49066487021205547009393310048, 6.10376163775855954087958455211, 6.80782312866686250662816999002, 7.65640957591422463077536474229, 9.661762965380425434721908562630, 10.24244330862356942790848010250, 11.41249333527498761606979552432, 11.79722324742835936439457124156