L(s) = 1 | − 2·3-s − 4·7-s + 9-s + 11-s + 4·13-s + 4·19-s + 8·21-s − 6·23-s + 4·27-s − 6·29-s − 8·31-s − 2·33-s − 2·37-s − 8·39-s + 6·41-s + 8·43-s + 6·47-s + 9·49-s + 6·53-s − 8·57-s + 12·59-s + 2·61-s − 4·63-s − 10·67-s + 12·69-s + 12·71-s + 16·73-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 1.51·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s + 0.917·19-s + 1.74·21-s − 1.25·23-s + 0.769·27-s − 1.11·29-s − 1.43·31-s − 0.348·33-s − 0.328·37-s − 1.28·39-s + 0.937·41-s + 1.21·43-s + 0.875·47-s + 9/7·49-s + 0.824·53-s − 1.05·57-s + 1.56·59-s + 0.256·61-s − 0.503·63-s − 1.22·67-s + 1.44·69-s + 1.42·71-s + 1.87·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−7.86499496632861719392953609421, −7.01609870378475161743947583743, −6.44241690116309043090441361867, −5.65889433188227421033845679887, −5.52958383648087111826963240906, −3.98244178628333179220455849882, −3.65149113231465708944230333700, −2.46535642408448141031036689969, −1.05453299331481824083824038720, 0,
1.05453299331481824083824038720, 2.46535642408448141031036689969, 3.65149113231465708944230333700, 3.98244178628333179220455849882, 5.52958383648087111826963240906, 5.65889433188227421033845679887, 6.44241690116309043090441361867, 7.01609870378475161743947583743, 7.86499496632861719392953609421