L(s) = 1 | − 4·5-s + 2·7-s − 3·9-s + 2·11-s − 13-s + 17-s + 4·23-s + 11·25-s + 2·29-s + 2·31-s − 8·35-s − 4·43-s + 12·45-s + 8·47-s − 3·49-s − 6·53-s − 8·55-s − 8·59-s − 2·61-s − 6·63-s + 4·65-s − 16·67-s + 14·71-s − 16·73-s + 4·77-s − 8·79-s + 9·81-s + ⋯ |
L(s) = 1 | − 1.78·5-s + 0.755·7-s − 9-s + 0.603·11-s − 0.277·13-s + 0.242·17-s + 0.834·23-s + 11/5·25-s + 0.371·29-s + 0.359·31-s − 1.35·35-s − 0.609·43-s + 1.78·45-s + 1.16·47-s − 3/7·49-s − 0.824·53-s − 1.07·55-s − 1.04·59-s − 0.256·61-s − 0.755·63-s + 0.496·65-s − 1.95·67-s + 1.66·71-s − 1.87·73-s + 0.455·77-s − 0.900·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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
−8.231611823260299394893519767719, −7.56075012227494814093389652188, −6.93135832561926676716833903383, −5.94560964015080227157758117432, −4.91084014676734747109483142034, −4.40411187841017741978139296451, −3.46444937113800303865713927742, −2.78761307496720857154270459339, −1.24541072403183618102061243025, 0,
1.24541072403183618102061243025, 2.78761307496720857154270459339, 3.46444937113800303865713927742, 4.40411187841017741978139296451, 4.91084014676734747109483142034, 5.94560964015080227157758117432, 6.93135832561926676716833903383, 7.56075012227494814093389652188, 8.231611823260299394893519767719