L(s) = 1 | + 2.94·3-s + 2.92i·5-s + i·7-s + 5.67·9-s + i·11-s + (2.34 − 2.74i)13-s + 8.62i·15-s + 4.73·17-s − 1.30i·19-s + 2.94i·21-s + 7.88·23-s − 3.58·25-s + 7.87·27-s + 6.34·29-s − 3.87i·31-s + ⋯ |
L(s) = 1 | + 1.70·3-s + 1.31i·5-s + 0.377i·7-s + 1.89·9-s + 0.301i·11-s + (0.649 − 0.760i)13-s + 2.22i·15-s + 1.14·17-s − 0.299i·19-s + 0.642i·21-s + 1.64·23-s − 0.716·25-s + 1.51·27-s + 1.17·29-s − 0.695i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.649 - 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.649 - 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.160390609\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.160390609\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 - iT \) |
| 13 | \( 1 + (-2.34 + 2.74i)T \) |
good | 3 | \( 1 - 2.94T + 3T^{2} \) |
| 5 | \( 1 - 2.92iT - 5T^{2} \) |
| 17 | \( 1 - 4.73T + 17T^{2} \) |
| 19 | \( 1 + 1.30iT - 19T^{2} \) |
| 23 | \( 1 - 7.88T + 23T^{2} \) |
| 29 | \( 1 - 6.34T + 29T^{2} \) |
| 31 | \( 1 + 3.87iT - 31T^{2} \) |
| 37 | \( 1 + 3.71iT - 37T^{2} \) |
| 41 | \( 1 + 7.91iT - 41T^{2} \) |
| 43 | \( 1 + 11.9T + 43T^{2} \) |
| 47 | \( 1 - 3.67iT - 47T^{2} \) |
| 53 | \( 1 + 9.14T + 53T^{2} \) |
| 59 | \( 1 - 14.1iT - 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 - 13.5iT - 67T^{2} \) |
| 71 | \( 1 + 8.37iT - 71T^{2} \) |
| 73 | \( 1 + 3.65iT - 73T^{2} \) |
| 79 | \( 1 + 8.06T + 79T^{2} \) |
| 83 | \( 1 - 13.7iT - 83T^{2} \) |
| 89 | \( 1 + 13.2iT - 89T^{2} \) |
| 97 | \( 1 - 9.09iT - 97T^{2} \) |
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
−8.582623208462415822517991203922, −7.77108759116770162041089386077, −7.31850246556246219534205135114, −6.59470523901460923521613376991, −5.66454265826383786431113124298, −4.57307861650835613412569511453, −3.46145328632918251427453124448, −3.06751288313812699696281647515, −2.51871670575625754091072191943, −1.34693122574795978561922816623,
1.14656354473231015302755231794, 1.64269353142940256525847350030, 3.13723401107902693403148472523, 3.42297668467640162668086528272, 4.61651283156277685589615038122, 4.94300584534262709395975189741, 6.29554008029134149589490140459, 7.07029233707998334524912239109, 8.042118175524870719927002072647, 8.333368156974550037063973905177