L(s) = 1 | − i·2-s + (−0.707 − 0.707i)3-s − 4-s + (−2.19 − 0.407i)5-s + (−0.707 + 0.707i)6-s + (1.64 + 1.64i)7-s + i·8-s + 1.00i·9-s + (−0.407 + 2.19i)10-s − 5.09i·11-s + (0.707 + 0.707i)12-s + 5.90i·13-s + (1.64 − 1.64i)14-s + (1.26 + 1.84i)15-s + 16-s + 0.720·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (−0.408 − 0.408i)3-s − 0.5·4-s + (−0.983 − 0.182i)5-s + (−0.288 + 0.288i)6-s + (0.621 + 0.621i)7-s + 0.353i·8-s + 0.333i·9-s + (−0.128 + 0.695i)10-s − 1.53i·11-s + (0.204 + 0.204i)12-s + 1.63i·13-s + (0.439 − 0.439i)14-s + (0.327 + 0.475i)15-s + 0.250·16-s + 0.174·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.338 + 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.338 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.140770011\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.140770011\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (2.19 + 0.407i)T \) |
| 37 | \( 1 + (-6.04 - 0.654i)T \) |
good | 7 | \( 1 + (-1.64 - 1.64i)T + 7iT^{2} \) |
| 11 | \( 1 + 5.09iT - 11T^{2} \) |
| 13 | \( 1 - 5.90iT - 13T^{2} \) |
| 17 | \( 1 - 0.720T + 17T^{2} \) |
| 19 | \( 1 + (-5.20 - 5.20i)T + 19iT^{2} \) |
| 23 | \( 1 + 2.47iT - 23T^{2} \) |
| 29 | \( 1 + (-2.09 + 2.09i)T - 29iT^{2} \) |
| 31 | \( 1 + (6.99 + 6.99i)T + 31iT^{2} \) |
| 41 | \( 1 - 6.58iT - 41T^{2} \) |
| 43 | \( 1 + 10.4iT - 43T^{2} \) |
| 47 | \( 1 + (-6.69 - 6.69i)T + 47iT^{2} \) |
| 53 | \( 1 + (-6.02 + 6.02i)T - 53iT^{2} \) |
| 59 | \( 1 + (-1.31 - 1.31i)T + 59iT^{2} \) |
| 61 | \( 1 + (0.136 + 0.136i)T + 61iT^{2} \) |
| 67 | \( 1 + (-8.82 + 8.82i)T - 67iT^{2} \) |
| 71 | \( 1 - 0.630T + 71T^{2} \) |
| 73 | \( 1 + (0.800 + 0.800i)T + 73iT^{2} \) |
| 79 | \( 1 + (1.35 + 1.35i)T + 79iT^{2} \) |
| 83 | \( 1 + (-0.0877 + 0.0877i)T - 83iT^{2} \) |
| 89 | \( 1 + (0.651 - 0.651i)T - 89iT^{2} \) |
| 97 | \( 1 + 5.99T + 97T^{2} \) |
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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
−9.657042418767450827398974716516, −8.782930004452520568117088835294, −8.176198443271464357709454326555, −7.38531428116214734701113395548, −6.14045203675691910334728545810, −5.36404037606424219872042940968, −4.28412304594394073141042947939, −3.45621403519220047305734730478, −2.10316294003457326203358454050, −0.821385669746681824531597731291,
0.882598759820739996538064417240, 3.05541179276204492861213695890, 4.12120771184491357683220967595, 4.91055404573278215481306800307, 5.52348478310530305086754219582, 7.14373136269497788831922744907, 7.25977440430436385160831937206, 8.104670184583189249531126765249, 9.139911682670902000671767626698, 10.06133007771189880588460562385