L(s) = 1 | − 2.46i·5-s + 5.48i·7-s − 10.5·11-s − 8.96i·13-s − 16.2·17-s − 2.96·19-s − 1.46i·23-s + 18.9·25-s − 25.0i·29-s + 10.5i·31-s + 13.5·35-s − 16.9i·37-s + 29.0·41-s + 34.9·43-s + 86.1i·47-s + ⋯ |
L(s) = 1 | − 0.492i·5-s + 0.783i·7-s − 0.962·11-s − 0.689i·13-s − 0.955·17-s − 0.156·19-s − 0.0635i·23-s + 0.757·25-s − 0.865i·29-s + 0.339i·31-s + 0.385·35-s − 0.458i·37-s + 0.707·41-s + 0.813·43-s + 1.83i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.456870114\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.456870114\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 2.46iT - 25T^{2} \) |
| 7 | \( 1 - 5.48iT - 49T^{2} \) |
| 11 | \( 1 + 10.5T + 121T^{2} \) |
| 13 | \( 1 + 8.96iT - 169T^{2} \) |
| 17 | \( 1 + 16.2T + 289T^{2} \) |
| 19 | \( 1 + 2.96T + 361T^{2} \) |
| 23 | \( 1 + 1.46iT - 529T^{2} \) |
| 29 | \( 1 + 25.0iT - 841T^{2} \) |
| 31 | \( 1 - 10.5iT - 961T^{2} \) |
| 37 | \( 1 + 16.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 29.0T + 1.68e3T^{2} \) |
| 43 | \( 1 - 34.9T + 1.84e3T^{2} \) |
| 47 | \( 1 - 86.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 80.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 66.4T + 3.48e3T^{2} \) |
| 61 | \( 1 - 0.966iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 113.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 90.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 51.7T + 5.32e3T^{2} \) |
| 79 | \( 1 - 80.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 79.9T + 6.88e3T^{2} \) |
| 89 | \( 1 - 142.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 45.8T + 9.40e3T^{2} \) |
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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
−9.042763628223413903755880212569, −8.101195250104184663443371596619, −7.62307026811957590962898019912, −6.44975783859226121647033282153, −5.73465654423805516110868268622, −4.99528326807207187215176515196, −4.23112727547306089156195679933, −2.89866153606556323683837545820, −2.27272910413048693885637413681, −0.816432090833609849843571168446,
0.45966142290642423231133171692, 1.91326545978170451085465371781, 2.85695673754211737705326189626, 3.87330936313697086856138506370, 4.67268319609312193287224783286, 5.54125688608137527964096943163, 6.70798466890330280277216150709, 6.99969794555827763521199020005, 7.920998338537408605538115671873, 8.700642890226027257690617866184