L(s) = 1 | − 2-s + (−0.707 + 0.707i)3-s + 4-s + (−2.23 + 0.0484i)5-s + (0.707 − 0.707i)6-s + (2.77 − 2.77i)7-s − 8-s − 1.00i·9-s + (2.23 − 0.0484i)10-s − 2.74i·11-s + (−0.707 + 0.707i)12-s − 2.09·13-s + (−2.77 + 2.77i)14-s + (1.54 − 1.61i)15-s + 16-s + 4.54i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.408 + 0.408i)3-s + 0.5·4-s + (−0.999 + 0.0216i)5-s + (0.288 − 0.288i)6-s + (1.04 − 1.04i)7-s − 0.353·8-s − 0.333i·9-s + (0.706 − 0.0153i)10-s − 0.826i·11-s + (−0.204 + 0.204i)12-s − 0.581·13-s + (−0.742 + 0.742i)14-s + (0.399 − 0.417i)15-s + 0.250·16-s + 1.10i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1003039788\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1003039788\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (2.23 - 0.0484i)T \) |
| 37 | \( 1 + (-3.14 - 5.20i)T \) |
good | 7 | \( 1 + (-2.77 + 2.77i)T - 7iT^{2} \) |
| 11 | \( 1 + 2.74iT - 11T^{2} \) |
| 13 | \( 1 + 2.09T + 13T^{2} \) |
| 17 | \( 1 - 4.54iT - 17T^{2} \) |
| 19 | \( 1 + (3.00 + 3.00i)T + 19iT^{2} \) |
| 23 | \( 1 + 1.09T + 23T^{2} \) |
| 29 | \( 1 + (3.41 - 3.41i)T - 29iT^{2} \) |
| 31 | \( 1 + (0.444 + 0.444i)T + 31iT^{2} \) |
| 41 | \( 1 - 3.49iT - 41T^{2} \) |
| 43 | \( 1 + 2.64T + 43T^{2} \) |
| 47 | \( 1 + (4.28 - 4.28i)T - 47iT^{2} \) |
| 53 | \( 1 + (4.10 + 4.10i)T + 53iT^{2} \) |
| 59 | \( 1 + (10.1 + 10.1i)T + 59iT^{2} \) |
| 61 | \( 1 + (4.77 + 4.77i)T + 61iT^{2} \) |
| 67 | \( 1 + (3.42 + 3.42i)T + 67iT^{2} \) |
| 71 | \( 1 - 0.512T + 71T^{2} \) |
| 73 | \( 1 + (9.57 - 9.57i)T - 73iT^{2} \) |
| 79 | \( 1 + (4.84 + 4.84i)T + 79iT^{2} \) |
| 83 | \( 1 + (-4.09 - 4.09i)T + 83iT^{2} \) |
| 89 | \( 1 + (9.59 - 9.59i)T - 89iT^{2} \) |
| 97 | \( 1 - 5.08iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.481599874312167817716777243653, −8.367664229579874170311619535469, −8.023365390803168424345254920258, −7.11886325153072384945614603996, −6.25483600060800934507642717847, −4.92965928880641262386808064452, −4.22265432136236109854221585532, −3.21257797818857810395071689573, −1.43619314761901943749157815676, −0.06086786063625462514964118252,
1.69507024233232380349483699140, 2.66424224895777219617023441361, 4.30395063488006702007135516098, 5.13289814621004619543990411481, 6.11846620535516535216418045549, 7.38113504957230419685437140947, 7.60730833733909286096201605163, 8.547963291116465815626733714879, 9.247814675843724395482420933899, 10.29468124653454927206801439239