L(s) = 1 | − 2.16·3-s + 8.18i·7-s − 4.29·9-s + 3.30·11-s + 8.86·13-s + 10.2i·17-s + (−18.5 − 4.03i)19-s − 17.7i·21-s − 0.896i·23-s + 28.8·27-s + 17.9i·29-s + 40.8i·31-s − 7.16·33-s − 7.68·37-s − 19.2·39-s + ⋯ |
L(s) = 1 | − 0.723·3-s + 1.16i·7-s − 0.477·9-s + 0.300·11-s + 0.681·13-s + 0.602i·17-s + (−0.977 − 0.212i)19-s − 0.845i·21-s − 0.0389i·23-s + 1.06·27-s + 0.617i·29-s + 1.31i·31-s − 0.217·33-s − 0.207·37-s − 0.492·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.247i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.968 + 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3699355340\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3699355340\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (18.5 + 4.03i)T \) |
good | 3 | \( 1 + 2.16T + 9T^{2} \) |
| 7 | \( 1 - 8.18iT - 49T^{2} \) |
| 11 | \( 1 - 3.30T + 121T^{2} \) |
| 13 | \( 1 - 8.86T + 169T^{2} \) |
| 17 | \( 1 - 10.2iT - 289T^{2} \) |
| 23 | \( 1 + 0.896iT - 529T^{2} \) |
| 29 | \( 1 - 17.9iT - 841T^{2} \) |
| 31 | \( 1 - 40.8iT - 961T^{2} \) |
| 37 | \( 1 + 7.68T + 1.36e3T^{2} \) |
| 41 | \( 1 - 69.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 25.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 12.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 55.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + 105. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 48.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 14.6T + 4.48e3T^{2} \) |
| 71 | \( 1 - 2.89iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 75.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 86.1iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 52.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 39.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 172.T + 9.40e3T^{2} \) |
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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
−9.263945695502873946517024701278, −8.661288324023572081217905964105, −8.154568516497927387934983570594, −6.73503327843255533006488624079, −6.26779002395113186877377768395, −5.50831170954017018892851127137, −4.80551818635053944118774001514, −3.60612224255548341910815932279, −2.59701682176847454845358056040, −1.43203749973932901059864287894,
0.12047912879820135532771622162, 1.07824769865890791277794244692, 2.51491649931538184542073315299, 3.81873499358465531301290892890, 4.37847463388070490439020181038, 5.51912799863523025135132946388, 6.18040525271381402530012557656, 6.94928073291881460217338384905, 7.73663909158252872455537608923, 8.615709051675246450730743356282