L(s) = 1 | − 2-s − i·3-s + 4-s + (−0.718 − 2.11i)5-s + i·6-s + 2.74i·7-s − 8-s − 9-s + (0.718 + 2.11i)10-s − 0.572·11-s − i·12-s + 3.04·13-s − 2.74i·14-s + (−2.11 + 0.718i)15-s + 16-s − 4.95·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577i·3-s + 0.5·4-s + (−0.321 − 0.946i)5-s + 0.408i·6-s + 1.03i·7-s − 0.353·8-s − 0.333·9-s + (0.227 + 0.669i)10-s − 0.172·11-s − 0.288i·12-s + 0.843·13-s − 0.734i·14-s + (−0.546 + 0.185i)15-s + 0.250·16-s − 1.20·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2282871826\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2282871826\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| 5 | \( 1 + (0.718 + 2.11i)T \) |
| 37 | \( 1 + (4.94 + 3.53i)T \) |
good | 7 | \( 1 - 2.74iT - 7T^{2} \) |
| 11 | \( 1 + 0.572T + 11T^{2} \) |
| 13 | \( 1 - 3.04T + 13T^{2} \) |
| 17 | \( 1 + 4.95T + 17T^{2} \) |
| 19 | \( 1 + 3.74iT - 19T^{2} \) |
| 23 | \( 1 + 2.98T + 23T^{2} \) |
| 29 | \( 1 + 7.36iT - 29T^{2} \) |
| 31 | \( 1 + 1.12iT - 31T^{2} \) |
| 41 | \( 1 + 11.7T + 41T^{2} \) |
| 43 | \( 1 + 6.91T + 43T^{2} \) |
| 47 | \( 1 - 0.776iT - 47T^{2} \) |
| 53 | \( 1 - 3.55iT - 53T^{2} \) |
| 59 | \( 1 - 9.27iT - 59T^{2} \) |
| 61 | \( 1 - 11.2iT - 61T^{2} \) |
| 67 | \( 1 - 9.09iT - 67T^{2} \) |
| 71 | \( 1 - 6.26T + 71T^{2} \) |
| 73 | \( 1 + 7.83iT - 73T^{2} \) |
| 79 | \( 1 - 0.716iT - 79T^{2} \) |
| 83 | \( 1 + 2.01iT - 83T^{2} \) |
| 89 | \( 1 + 8.33iT - 89T^{2} \) |
| 97 | \( 1 + 15.6T + 97T^{2} \) |
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
−8.977617029981689947920712977870, −8.706086600892706283936038857553, −8.017480471111603820214686418731, −6.98250446406001630964538551374, −6.09227115477346642468329613313, −5.28874650687949696200232825804, −4.09179012033753307252046319256, −2.61920104374677144446001166966, −1.63300061875397157625027439305, −0.12226112838232024312894820104,
1.78778171941287726819177382501, 3.31200576399057514474476771070, 3.86186675675385282117083199880, 5.13778261452997680093397432091, 6.54651801748300419426298004076, 6.83423609463270721086308031106, 8.021185708958286593412215047004, 8.523241225177086196751078824359, 9.693093091691514218147124814051, 10.35259000295273534843219266440