L(s) = 1 | − 2.23·3-s + (1.58 + 1.58i)5-s + 2.82i·7-s + 2.00·9-s + 5i·13-s + (−3.53 − 3.53i)15-s − 3.16·17-s − 7.07·19-s − 6.32i·21-s + (−2.23 + 4.24i)23-s + 5.00i·25-s + 2.23·27-s + 9·29-s − 6.70i·31-s + (−4.47 + 4.47i)35-s + ⋯ |
L(s) = 1 | − 1.29·3-s + (0.707 + 0.707i)5-s + 1.06i·7-s + 0.666·9-s + 1.38i·13-s + (−0.912 − 0.912i)15-s − 0.766·17-s − 1.62·19-s − 1.38i·21-s + (−0.466 + 0.884i)23-s + 1.00i·25-s + 0.430·27-s + 1.67·29-s − 1.20i·31-s + (−0.755 + 0.755i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 + 0.295i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.955 + 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4798457647\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4798457647\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.58 - 1.58i)T \) |
| 23 | \( 1 + (2.23 - 4.24i)T \) |
good | 3 | \( 1 + 2.23T + 3T^{2} \) |
| 7 | \( 1 - 2.82iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 5iT - 13T^{2} \) |
| 17 | \( 1 + 3.16T + 17T^{2} \) |
| 19 | \( 1 + 7.07T + 19T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 + 6.70iT - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 + 1.41iT - 43T^{2} \) |
| 47 | \( 1 + 6.70T + 47T^{2} \) |
| 53 | \( 1 - 6.32T + 53T^{2} \) |
| 59 | \( 1 - 4.47iT - 59T^{2} \) |
| 61 | \( 1 + 9.48iT - 61T^{2} \) |
| 67 | \( 1 - 2.82iT - 67T^{2} \) |
| 71 | \( 1 + 6.70iT - 71T^{2} \) |
| 73 | \( 1 + 15iT - 73T^{2} \) |
| 79 | \( 1 - 7.07T + 79T^{2} \) |
| 83 | \( 1 - 12.7iT - 83T^{2} \) |
| 89 | \( 1 - 3.16iT - 89T^{2} \) |
| 97 | \( 1 + 18.9T + 97T^{2} \) |
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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
−9.708967219173620118150218080574, −9.049791829627991406491505804760, −8.195058346188083675250475798023, −6.74387214640406637848530674069, −6.49779543061745326663392832851, −5.83707266137454959840282645989, −4.99806623090416655144299468158, −4.09921021996700728511168770502, −2.57003286356525309209527726419, −1.82285683627161372738462519305,
0.22395134546757526020397620197, 1.17682978093125267313057398423, 2.64141457543551992901260100630, 4.19179623249361515355106448217, 4.79880230665090811427750376781, 5.57000197065781178709376226738, 6.44253460894329549783509701014, 6.83637972410096743091788117291, 8.255287332618515851577130560180, 8.612184565786639590366266451465