L(s) = 1 | + i·2-s − i·3-s − 4-s + (−2.23 + 0.0526i)5-s + 6-s + 2i·7-s − i·8-s − 9-s + (−0.0526 − 2.23i)10-s − 0.470·11-s + i·12-s − 6.47i·13-s − 2·14-s + (0.0526 + 2.23i)15-s + 16-s + 7.04i·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (−0.999 + 0.0235i)5-s + 0.408·6-s + 0.755i·7-s − 0.353i·8-s − 0.333·9-s + (−0.0166 − 0.706i)10-s − 0.141·11-s + 0.288i·12-s − 1.79i·13-s − 0.534·14-s + (0.0135 + 0.577i)15-s + 0.250·16-s + 1.70i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0235i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.15891 + 0.0136488i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.15891 + 0.0136488i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| 5 | \( 1 + (2.23 - 0.0526i)T \) |
| 31 | \( 1 + T \) |
good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 0.470T + 11T^{2} \) |
| 13 | \( 1 + 6.47iT - 13T^{2} \) |
| 17 | \( 1 - 7.04iT - 17T^{2} \) |
| 19 | \( 1 - 7.04T + 19T^{2} \) |
| 23 | \( 1 + 6.94iT - 23T^{2} \) |
| 29 | \( 1 - 6.94T + 29T^{2} \) |
| 37 | \( 1 + 1.78iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 0.210iT - 43T^{2} \) |
| 47 | \( 1 + 7.04iT - 47T^{2} \) |
| 53 | \( 1 - 3.15iT - 53T^{2} \) |
| 59 | \( 1 - 7.15T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 + 8.26iT - 67T^{2} \) |
| 71 | \( 1 - 0.260T + 71T^{2} \) |
| 73 | \( 1 + 11.8iT - 73T^{2} \) |
| 79 | \( 1 - 1.89T + 79T^{2} \) |
| 83 | \( 1 - 11.7iT - 83T^{2} \) |
| 89 | \( 1 + 12.0T + 89T^{2} \) |
| 97 | \( 1 + 3.52iT - 97T^{2} \) |
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
−10.12461864792402564012301247190, −8.749850019418616354169742231637, −8.230007075327193991581821151282, −7.70603399963008599449560404552, −6.71097129984658726387873078418, −5.79443270245837435101869813423, −5.05774721293566686167443644511, −3.72840688322061401172338255051, −2.73154872617466493573894720637, −0.75028415678710255794022844439,
1.01461484730051513890731435765, 2.82763970041976481777680247682, 3.76122966367400274574582673828, 4.50416628739632516664040818881, 5.28579331807154264471005186648, 6.97300332411234527161313936242, 7.46318202480501323994333836035, 8.616980038011703230432543684148, 9.489031096826429258656131482698, 9.932713248122373518705156459661