L(s) = 1 | + (−2.91 − 0.690i)3-s − 2.77i·5-s + 1.17·7-s + (8.04 + 4.02i)9-s − 8.93i·11-s − 10.2·13-s + (−1.91 + 8.10i)15-s + 21.1i·17-s − 27.2·19-s + (−3.42 − 0.809i)21-s + 4.79i·23-s + 17.3·25-s + (−20.7 − 17.3i)27-s − 9.80i·29-s + 32.5·31-s + ⋯ |
L(s) = 1 | + (−0.973 − 0.230i)3-s − 0.554i·5-s + 0.167·7-s + (0.894 + 0.447i)9-s − 0.812i·11-s − 0.792·13-s + (−0.127 + 0.540i)15-s + 1.24i·17-s − 1.43·19-s + (−0.163 − 0.0385i)21-s + 0.208i·23-s + 0.692·25-s + (−0.767 − 0.641i)27-s − 0.338i·29-s + 1.04·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 - 0.973i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6354825819\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6354825819\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.91 + 0.690i)T \) |
| 23 | \( 1 - 4.79iT \) |
good | 5 | \( 1 + 2.77iT - 25T^{2} \) |
| 7 | \( 1 - 1.17T + 49T^{2} \) |
| 11 | \( 1 + 8.93iT - 121T^{2} \) |
| 13 | \( 1 + 10.2T + 169T^{2} \) |
| 17 | \( 1 - 21.1iT - 289T^{2} \) |
| 19 | \( 1 + 27.2T + 361T^{2} \) |
| 29 | \( 1 + 9.80iT - 841T^{2} \) |
| 31 | \( 1 - 32.5T + 961T^{2} \) |
| 37 | \( 1 + 46.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + 59.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 10.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + 21.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 75.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 65.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 29.5T + 3.72e3T^{2} \) |
| 67 | \( 1 - 38.5T + 4.48e3T^{2} \) |
| 71 | \( 1 - 138. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 73.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 28.2T + 6.24e3T^{2} \) |
| 83 | \( 1 - 119. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 115. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 116.T + 9.40e3T^{2} \) |
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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
−10.13157183683353889064756217323, −8.830683913368923643622775906612, −8.288391094854344733955290238166, −7.22095310892885349740323552692, −6.35245855114460325012600274802, −5.61848369091942648141167979763, −4.74685336410028777490726378997, −3.89941453559928603318161307907, −2.24701584353636501452708698733, −1.01514665552454293409630622034,
0.25874951818207139088805252970, 1.92502616068428499230559016498, 3.16672052956250207141762682640, 4.68320143051789536945713771056, 4.85187809501473919064672378405, 6.26715654828817422610853716552, 6.83945143480425573957542325752, 7.56438161430989270130084499443, 8.761125384465889159284415058258, 9.829018926952496005090586195409