L(s) = 1 | + (1.87 + 1.87i)3-s + (1.41 + 1.41i)7-s + 4i·9-s + 5.29i·11-s + (1.87 + 1.87i)13-s − 5.29·19-s + 5.29i·21-s + (−4.69 + 0.957i)23-s + (−1.87 + 1.87i)27-s − i·29-s + 3·31-s + (−9.89 + 9.89i)33-s + (−7.07 − 7.07i)37-s + 7i·39-s + 9·41-s + ⋯ |
L(s) = 1 | + (1.08 + 1.08i)3-s + (0.534 + 0.534i)7-s + 1.33i·9-s + 1.59i·11-s + (0.518 + 0.518i)13-s − 1.21·19-s + 1.15i·21-s + (−0.979 + 0.199i)23-s + (−0.360 + 0.360i)27-s − 0.185i·29-s + 0.538·31-s + (−1.72 + 1.72i)33-s + (−1.16 − 1.16i)37-s + 1.12i·39-s + 1.40·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.684 - 0.728i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.684 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.555521999\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.555521999\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + (4.69 - 0.957i)T \) |
good | 3 | \( 1 + (-1.87 - 1.87i)T + 3iT^{2} \) |
| 7 | \( 1 + (-1.41 - 1.41i)T + 7iT^{2} \) |
| 11 | \( 1 - 5.29iT - 11T^{2} \) |
| 13 | \( 1 + (-1.87 - 1.87i)T + 13iT^{2} \) |
| 17 | \( 1 + 17iT^{2} \) |
| 19 | \( 1 + 5.29T + 19T^{2} \) |
| 29 | \( 1 + iT - 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 + (7.07 + 7.07i)T + 37iT^{2} \) |
| 41 | \( 1 - 9T + 41T^{2} \) |
| 43 | \( 1 + (-4.24 + 4.24i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.87 - 1.87i)T - 47iT^{2} \) |
| 53 | \( 1 + (-2.82 + 2.82i)T - 53iT^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 - 5.29iT - 61T^{2} \) |
| 67 | \( 1 + (1.41 + 1.41i)T + 67iT^{2} \) |
| 71 | \( 1 + 9T + 71T^{2} \) |
| 73 | \( 1 + (-9.35 - 9.35i)T + 73iT^{2} \) |
| 79 | \( 1 + 5.29T + 79T^{2} \) |
| 83 | \( 1 + (8.48 - 8.48i)T - 83iT^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 + (1.41 + 1.41i)T + 97iT^{2} \) |
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
−9.138221406362970864803056685048, −8.745069462124014205467217378528, −7.971091366227148392795735739341, −7.16968567307917642555854308753, −6.11026832502704674245329844741, −5.05789389923897759321455822512, −4.24877147272939716674832131310, −3.85562517242067867961237719764, −2.43756008137031246465232122615, −1.94779603550709466092332692891,
0.74085678562039001911621277070, 1.73700943139679501446163818271, 2.81289124091358647711167890323, 3.55235072736731173064652357326, 4.54617663490459499919925485466, 5.91301557551477800307827317120, 6.40092561454449376150284926726, 7.41789094354718374191710682202, 8.102861633180929075471667677838, 8.450974661986262297217955851904