| L(s) = 1 | − 0.638·3-s + 9.06i·7-s − 8.59·9-s + (10.0 − 4.36i)11-s − 10.4i·13-s − 3.39i·17-s − 28.0i·19-s − 5.79i·21-s + 18.8·23-s + 11.2·27-s + 22.2i·29-s + 0.00744·31-s + (−6.44 + 2.79i)33-s + 8.96·37-s + 6.66i·39-s + ⋯ |
| L(s) = 1 | − 0.212·3-s + 1.29i·7-s − 0.954·9-s + (0.917 − 0.397i)11-s − 0.802i·13-s − 0.199i·17-s − 1.47i·19-s − 0.275i·21-s + 0.818·23-s + 0.416·27-s + 0.766i·29-s + 0.000240·31-s + (−0.195 + 0.0845i)33-s + 0.242·37-s + 0.170i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.653634608\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.653634608\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (-10.0 + 4.36i)T \) |
| good | 3 | \( 1 + 0.638T + 9T^{2} \) |
| 7 | \( 1 - 9.06iT - 49T^{2} \) |
| 13 | \( 1 + 10.4iT - 169T^{2} \) |
| 17 | \( 1 + 3.39iT - 289T^{2} \) |
| 19 | \( 1 + 28.0iT - 361T^{2} \) |
| 23 | \( 1 - 18.8T + 529T^{2} \) |
| 29 | \( 1 - 22.2iT - 841T^{2} \) |
| 31 | \( 1 - 0.00744T + 961T^{2} \) |
| 37 | \( 1 - 8.96T + 1.36e3T^{2} \) |
| 41 | \( 1 - 16.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 81.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 59.1T + 2.20e3T^{2} \) |
| 53 | \( 1 - 60.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 81.5T + 3.48e3T^{2} \) |
| 61 | \( 1 - 114. iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 99.1T + 4.48e3T^{2} \) |
| 71 | \( 1 - 70.3T + 5.04e3T^{2} \) |
| 73 | \( 1 - 16.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 118. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 30.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 4.02T + 7.92e3T^{2} \) |
| 97 | \( 1 - 101.T + 9.40e3T^{2} \) |
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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
−9.411522611336065453012801695167, −8.936846149940039783114072441553, −8.320215084379116112268057665063, −7.12553387122249677013696200820, −6.17166349785376812639451644426, −5.54184535936835342125973357506, −4.70284151706085378408447250242, −3.17747251794445501565772659547, −2.55713006443585704758828502476, −0.854938718001899512097713819010,
0.74420020449274122274774176951, 2.02348265532834509376982584028, 3.62036012493213035849257179931, 4.16052174423626105601928243951, 5.34941105013810661619091920641, 6.35011023242158523486435940752, 7.03004834462474126963134073452, 7.904838123589371609312895014072, 8.834175185777383331178377921796, 9.634422495388290549163362981036
