L(s) = 1 | + 1.73i·5-s + (2 − 1.73i)7-s + 3.46i·11-s − 3.46i·13-s + 1.73i·17-s − 2·19-s + 2.00·25-s + 6·29-s + 2·31-s + (2.99 + 3.46i)35-s + 37-s + 5.19i·41-s + 1.73i·43-s − 3·47-s + (1.00 − 6.92i)49-s + ⋯ |
L(s) = 1 | + 0.774i·5-s + (0.755 − 0.654i)7-s + 1.04i·11-s − 0.960i·13-s + 0.420i·17-s − 0.458·19-s + 0.400·25-s + 1.11·29-s + 0.359·31-s + (0.507 + 0.585i)35-s + 0.164·37-s + 0.811i·41-s + 0.264i·43-s − 0.437·47-s + (0.142 − 0.989i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.011822455\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.011822455\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2 + 1.73i)T \) |
good | 5 | \( 1 - 1.73iT - 5T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 1.73iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 - 5.19iT - 41T^{2} \) |
| 43 | \( 1 - 1.73iT - 43T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 - 12T + 53T^{2} \) |
| 59 | \( 1 - 3T + 59T^{2} \) |
| 61 | \( 1 - 10.3iT - 61T^{2} \) |
| 67 | \( 1 - 10.3iT - 67T^{2} \) |
| 71 | \( 1 + 13.8iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 1.73iT - 79T^{2} \) |
| 83 | \( 1 + 9T + 83T^{2} \) |
| 89 | \( 1 - 6.92iT - 89T^{2} \) |
| 97 | \( 1 + 3.46iT - 97T^{2} \) |
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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
−8.596145739783728379344149036936, −8.034925581622747721991614425366, −7.24614113952464245359521577162, −6.73961271961020402772990681626, −5.79484833374628755163853077930, −4.80627462284735285995418671014, −4.20492015467822639094068765075, −3.12895080752269676522959116115, −2.24569575228573954634634820170, −1.05047054358958422598057850152,
0.76973689203848763629691502914, 1.89020282111464465092488994864, 2.87684242128108222421340472720, 4.07162418257711142966903336517, 4.81809910941579823533122901722, 5.47163910060977024061727019801, 6.30170367510470301474186080147, 7.12082006532309916508943525772, 8.213700704646330522583304468171, 8.615569725576451338330027133995