L(s) = 1 | − 2i·2-s + (−1.34 − 5.01i)3-s − 4·4-s + 4.58·5-s + (−10.0 + 2.69i)6-s − 31.2i·7-s + 8i·8-s + (−23.3 + 13.5i)9-s − 9.17i·10-s − 0.810·11-s + (5.38 + 20.0i)12-s − 14.3·13-s − 62.5·14-s + (−6.18 − 23.0i)15-s + 16·16-s − 45.4·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (−0.259 − 0.965i)3-s − 0.5·4-s + 0.410·5-s + (−0.682 + 0.183i)6-s − 1.68i·7-s + 0.353i·8-s + (−0.865 + 0.500i)9-s − 0.290i·10-s − 0.0222·11-s + (0.129 + 0.482i)12-s − 0.306·13-s − 1.19·14-s + (−0.106 − 0.396i)15-s + 0.250·16-s − 0.647·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.901 - 0.431i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.901 - 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.228229 + 1.00519i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.228229 + 1.00519i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2iT \) |
| 3 | \( 1 + (1.34 + 5.01i)T \) |
| 23 | \( 1 + (-83.7 - 71.7i)T \) |
good | 5 | \( 1 - 4.58T + 125T^{2} \) |
| 7 | \( 1 + 31.2iT - 343T^{2} \) |
| 11 | \( 1 + 0.810T + 1.33e3T^{2} \) |
| 13 | \( 1 + 14.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 45.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 11.6iT - 6.85e3T^{2} \) |
| 29 | \( 1 - 98.1iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 56.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 398. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 97.5iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 323. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 341. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 500.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 37.2iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 667. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 791. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 854. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 119.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 604. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 813.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 460.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 710. iT - 9.12e5T^{2} \) |
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
−12.18080589516278824073094711073, −11.05526211175798328993806380130, −10.40589180313961280760121727224, −9.099870189852099897043912916051, −7.65903696418976344724100150272, −6.85745320600792328038869115664, −5.32636323578458858261980205908, −3.76487926029836402177312407364, −1.95255233149210696757821774786, −0.52357550813331929175221369445,
2.71765389923765658609212326108, 4.59770596060659905144187822380, 5.58776914869556374680367971498, 6.43164050333151676400607274580, 8.296918897425750121741578012918, 9.140528435628506244014309076939, 9.885408585249066245400892727445, 11.27771482199113960646701151014, 12.22266340815279296583964347128, 13.41656084599584153079269457166