| L(s) = 1 | − i·3-s + 3i·7-s + 2·9-s + 2·11-s − i·13-s − 5i·17-s − 19-s + 3·21-s + i·23-s − 5i·27-s + 3·29-s + 4·31-s − 2i·33-s + 2i·37-s − 39-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 1.13i·7-s + 0.666·9-s + 0.603·11-s − 0.277i·13-s − 1.21i·17-s − 0.229·19-s + 0.654·21-s + 0.208i·23-s − 0.962i·27-s + 0.557·29-s + 0.718·31-s − 0.348i·33-s + 0.328i·37-s − 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.131214705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.131214705\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + iT - 3T^{2} \) |
| 7 | \( 1 - 3iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 + 5iT - 17T^{2} \) |
| 23 | \( 1 - iT - 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 9iT - 53T^{2} \) |
| 59 | \( 1 + T + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 - 13iT - 67T^{2} \) |
| 71 | \( 1 - 10T + 71T^{2} \) |
| 73 | \( 1 + 9iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 10iT - 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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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
−8.380519276914765202130062799307, −7.77434848980775936382310046910, −6.72798501579055834463631913508, −6.53625648618392145621412596375, −5.39679155135859262068134042788, −4.82924182779664761793597756632, −3.74311036894514553072594356463, −2.72860112158830787645053082073, −1.94866917095304004562577190739, −0.824261714270010477597282677940,
0.943079307186390359816593907397, 1.92830454358131128596931199848, 3.36063433232076111183966528523, 4.09453531427954223741771908931, 4.44301408788742981283387886075, 5.48266750040847014794515410383, 6.59521731372866378725017474374, 6.88750124009191480934460013894, 7.87812407280192587973044173695, 8.528025274072942944335659489349
