| L(s) = 1 | − 2.23·5-s + 5.23i·7-s + 7.70i·23-s + 5.00·25-s + 6·29-s − 11.7i·35-s − 4.47·41-s + 6.76i·43-s − 0.291i·47-s − 20.4·49-s − 13.4·61-s − 14.1i·67-s + 4.29i·83-s + 6·89-s − 18·101-s + ⋯ |
| L(s) = 1 | − 0.999·5-s + 1.97i·7-s + 1.60i·23-s + 1.00·25-s + 1.11·29-s − 1.97i·35-s − 0.698·41-s + 1.03i·43-s − 0.0425i·47-s − 2.91·49-s − 1.71·61-s − 1.73i·67-s + 0.471i·83-s + 0.635·89-s − 1.79·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6952770076\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6952770076\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2.23T \) |
| good | 7 | \( 1 - 5.23iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 7.70iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 - 6.76iT - 43T^{2} \) |
| 47 | \( 1 + 0.291iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + 14.1iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 4.29iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.156044629995084048698952006352, −8.277699984630688528941804473611, −7.917487583153718553335101523206, −6.85569351824268536325027366976, −6.05177459914652225909930945346, −5.28511163388481562689631149548, −4.58162823645063976994201416955, −3.38336670949277615413446009108, −2.77114662479300092950477582243, −1.58463873544645348315168979470,
0.24662495179104622328045853099, 1.18549358138223781238139831968, 2.81803368090228004035418559472, 3.78278068563295678603713041699, 4.32200934656417424501909697701, 4.98624215570067044424131924143, 6.46203456046761212289197510314, 6.92175869773088652962492326188, 7.64765011637233389321377239522, 8.226004794726576829263397363342
