| L(s) = 1 | + (0.144 + 1.40i)2-s + (−1.95 + 0.406i)4-s + 3.62i·7-s + (−0.855 − 2.69i)8-s + 6.20i·11-s − 0.578·13-s + (−5.10 + 0.524i)14-s + (3.66 − 1.59i)16-s + 1.42i·17-s − 5.62i·19-s + (−8.72 + 0.897i)22-s + 5.62i·23-s + (−0.0836 − 0.813i)26-s + (−1.47 − 7.10i)28-s − 2i·29-s + ⋯ |
| L(s) = 1 | + (0.102 + 0.994i)2-s + (−0.979 + 0.203i)4-s + 1.37i·7-s + (−0.302 − 0.953i)8-s + 1.87i·11-s − 0.160·13-s + (−1.36 + 0.140i)14-s + (0.917 − 0.398i)16-s + 0.344i·17-s − 1.29i·19-s + (−1.86 + 0.191i)22-s + 1.17i·23-s + (−0.0163 − 0.159i)26-s + (−0.278 − 1.34i)28-s − 0.371i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.717 + 0.696i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.717 + 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9831767839\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9831767839\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.144 - 1.40i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 3.62iT - 7T^{2} \) |
| 11 | \( 1 - 6.20iT - 11T^{2} \) |
| 13 | \( 1 + 0.578T + 13T^{2} \) |
| 17 | \( 1 - 1.42iT - 17T^{2} \) |
| 19 | \( 1 + 5.62iT - 19T^{2} \) |
| 23 | \( 1 - 5.62iT - 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 + 2.57T + 31T^{2} \) |
| 37 | \( 1 - 7.83T + 37T^{2} \) |
| 41 | \( 1 + 5.25T + 41T^{2} \) |
| 43 | \( 1 + 7.25T + 43T^{2} \) |
| 47 | \( 1 - 6.78iT - 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 2.20iT - 59T^{2} \) |
| 61 | \( 1 + 12.4iT - 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 8.41T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 5.42T + 79T^{2} \) |
| 83 | \( 1 + 3.25T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 + 4.84iT - 97T^{2} \) |
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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
−9.451023130728991348153847878529, −9.039882707008723433855806037100, −8.064897204935945213100283026692, −7.34747895173629587604792148459, −6.63521799522264486053748056536, −5.74646780070225912208540375171, −4.98939186888883080408305205977, −4.35055993662486663772933331049, −3.00500960428827188681776524827, −1.85698537608781814113191617839,
0.37148939753864363778571737355, 1.36867600292601473460226911930, 2.86033059454761547769106175297, 3.64732608536314874955426216030, 4.32389639437147158973015393360, 5.41647279509762489078804555190, 6.22107418976081607985485446937, 7.31679554711097515315145795696, 8.309257122119896565419239077971, 8.710995909995994161548829844283
