| L(s) = 1 | + i·2-s − 4-s + 1.70i·7-s − i·8-s + 1.70·11-s − i·13-s − 1.70·14-s + 16-s + 5.70i·17-s − 4.70·19-s + 1.70i·22-s + 26-s − 1.70i·28-s + 6.40·29-s + 9.10·31-s + i·32-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.643i·7-s − 0.353i·8-s + 0.513·11-s − 0.277i·13-s − 0.454·14-s + 0.250·16-s + 1.38i·17-s − 1.07·19-s + 0.362i·22-s + 0.196·26-s − 0.321i·28-s + 1.18·29-s + 1.63·31-s + 0.176i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.437443015\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.437443015\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + iT \) |
| good | 7 | \( 1 - 1.70iT - 7T^{2} \) |
| 11 | \( 1 - 1.70T + 11T^{2} \) |
| 17 | \( 1 - 5.70iT - 17T^{2} \) |
| 19 | \( 1 + 4.70T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 6.40T + 29T^{2} \) |
| 31 | \( 1 - 9.10T + 31T^{2} \) |
| 37 | \( 1 + 4.70iT - 37T^{2} \) |
| 41 | \( 1 + 2.70T + 41T^{2} \) |
| 43 | \( 1 - 1.40iT - 43T^{2} \) |
| 47 | \( 1 - 7iT - 47T^{2} \) |
| 53 | \( 1 - 10.4iT - 53T^{2} \) |
| 59 | \( 1 + 3.70T + 59T^{2} \) |
| 61 | \( 1 + 5.10T + 61T^{2} \) |
| 67 | \( 1 + 6.40iT - 67T^{2} \) |
| 71 | \( 1 + 4.70T + 71T^{2} \) |
| 73 | \( 1 + 12iT - 73T^{2} \) |
| 79 | \( 1 + 0.701T + 79T^{2} \) |
| 83 | \( 1 - 4.29iT - 83T^{2} \) |
| 89 | \( 1 + 1.40T + 89T^{2} \) |
| 97 | \( 1 - 15.4iT - 97T^{2} \) |
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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
−8.311353571195392780725234288229, −7.85516538328857580197310743646, −6.83373480038600671653915133028, −6.14818877053540847696121274572, −5.90275190404228325288598588795, −4.72142197772678143095497659108, −4.26356039615854689577666172016, −3.24933086278471458165605094752, −2.27973483301547936810215092228, −1.13493562916460593838828628316,
0.41657384959657964327298468532, 1.37794086061822785142910042446, 2.47469667524298360828968110094, 3.18835277127837110534785680603, 4.22110443072731511381924566993, 4.58867183108736935619249509373, 5.49581999486699338154581736490, 6.64847780909361555251212622860, 6.87111103441240448618439103101, 7.986445277346041455289751722002
