L(s) = 1 | − 3i·3-s − 24i·7-s − 9·9-s − 52·11-s + 22i·13-s + 14i·17-s − 20·19-s − 72·21-s + 168i·23-s + 27i·27-s − 230·29-s + 288·31-s + 156i·33-s + 34i·37-s + 66·39-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 1.29i·7-s − 0.333·9-s − 1.42·11-s + 0.469i·13-s + 0.199i·17-s − 0.241·19-s − 0.748·21-s + 1.52i·23-s + 0.192i·27-s − 1.47·29-s + 1.66·31-s + 0.822i·33-s + 0.151i·37-s + 0.270·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.171041577\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.171041577\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 3iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 24iT - 343T^{2} \) |
| 11 | \( 1 + 52T + 1.33e3T^{2} \) |
| 13 | \( 1 - 22iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 14iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 20T + 6.85e3T^{2} \) |
| 23 | \( 1 - 168iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 230T + 2.43e4T^{2} \) |
| 31 | \( 1 - 288T + 2.97e4T^{2} \) |
| 37 | \( 1 - 34iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 122T + 6.89e4T^{2} \) |
| 43 | \( 1 - 188iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 256iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 338iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 100T + 2.05e5T^{2} \) |
| 61 | \( 1 - 742T + 2.26e5T^{2} \) |
| 67 | \( 1 + 84iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 328T + 3.57e5T^{2} \) |
| 73 | \( 1 + 38iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 240T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.21e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 330T + 7.04e5T^{2} \) |
| 97 | \( 1 + 866iT - 9.12e5T^{2} \) |
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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
−9.584102902399696338862671948576, −8.368261792575798144698269938489, −7.66087731357082742310527842421, −7.17261689491423870930504253545, −6.16942378039672201968295999020, −5.21621635503340887771324000148, −4.21508581911074986114216732787, −3.19720729963833811024921012866, −1.99882877623340155393399687571, −0.840521063463406905970192718478,
0.34723084631522976659882575486, 2.32942035284947921377474169413, 2.82584000500249093327824900980, 4.17439512547816117632999192290, 5.24526577752151801643046017886, 5.64059701560524747636056617506, 6.73126291108450120801948082546, 7.978729162394650041315246579078, 8.471467123230260455410486401530, 9.305199251133958269572880601264