| L(s) = 1 | + (−8.70 + 2.27i)3-s + 28.9i·5-s − 2.36·7-s + (70.6 − 39.6i)9-s − 21.1i·11-s − 259.·13-s + (−65.8 − 251. i)15-s + 350. i·17-s − 308.·19-s + (20.5 − 5.37i)21-s − 282. i·23-s − 211.·25-s + (−524. + 506. i)27-s + 1.23e3i·29-s − 158.·31-s + ⋯ |
| L(s) = 1 | + (−0.967 + 0.253i)3-s + 1.15i·5-s − 0.0481·7-s + (0.871 − 0.489i)9-s − 0.175i·11-s − 1.53·13-s + (−0.292 − 1.11i)15-s + 1.21i·17-s − 0.855·19-s + (0.0466 − 0.0121i)21-s − 0.534i·23-s − 0.339·25-s + (−0.719 + 0.694i)27-s + 1.47i·29-s − 0.164·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.253 + 0.967i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.253 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.3383265605\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3383265605\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (8.70 - 2.27i)T \) |
| good | 5 | \( 1 - 28.9iT - 625T^{2} \) |
| 7 | \( 1 + 2.36T + 2.40e3T^{2} \) |
| 11 | \( 1 + 21.1iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 259.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 350. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 308.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 282. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.23e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 158.T + 9.23e5T^{2} \) |
| 37 | \( 1 + 890.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 1.78e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 3.44e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 3.40e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 580. iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 3.75e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 205.T + 1.38e7T^{2} \) |
| 67 | \( 1 + 610.T + 2.01e7T^{2} \) |
| 71 | \( 1 - 3.01e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 5.23e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 6.54e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 1.30e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 5.44e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.87e3T + 8.85e7T^{2} \) |
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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
−10.55442915930969736031432450609, −10.03645401576641018884817890120, −8.765056017113564069151844598348, −7.33111092888631946031432600936, −6.73155524184153852743188021844, −5.77214796031682544607240076270, −4.65910743965440147891511181384, −3.46919598851992798510305511789, −2.06059587929288189242335816106, −0.13514333661870411565634922329,
0.920763005276119178550054183134, 2.35162027868593058545522127104, 4.43518600676595928843443761569, 4.96021251280906547462403203644, 5.98340702218027896933336577927, 7.16864194231442962404058938027, 7.942404073147182407769926354079, 9.306391158669562286903602178953, 9.869341601492610345748707366938, 11.08046625241168304606984157667
