| L(s) = 1 | − 2-s + 4-s + 1.41·5-s − 8-s + 9-s − 1.41·10-s + 16-s − 18-s + 1.41·20-s + 1.00·25-s − 1.41·29-s − 32-s + 36-s − 1.41·37-s − 1.41·40-s − 1.41·41-s + 1.41·45-s + 49-s − 1.00·50-s + 1.41·58-s + 1.41·61-s + 64-s − 72-s − 1.41·73-s + 1.41·74-s + 1.41·80-s + 81-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s + 1.41·5-s − 8-s + 9-s − 1.41·10-s + 16-s − 18-s + 1.41·20-s + 1.00·25-s − 1.41·29-s − 32-s + 36-s − 1.41·37-s − 1.41·40-s − 1.41·41-s + 1.41·45-s + 49-s − 1.00·50-s + 1.41·58-s + 1.41·61-s + 64-s − 72-s − 1.41·73-s + 1.41·74-s + 1.41·80-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9224402466\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9224402466\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - 1.41T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 1.41T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 1.41T + T^{2} \) |
| 41 | \( 1 + 1.41T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.41T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.41T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - 1.41T + T^{2} \) |
| 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
−10.07119591672476248371186517129, −9.253176209357900988401001777534, −8.616008782696301635635136454714, −7.44553303887263391257875235573, −6.83164465893778212385571895107, −5.95573941740973026735021173568, −5.14169447944175471979683650678, −3.61006690859387285815609488211, −2.23508563830816040785000016830, −1.49415910324576107748984374736,
1.49415910324576107748984374736, 2.23508563830816040785000016830, 3.61006690859387285815609488211, 5.14169447944175471979683650678, 5.95573941740973026735021173568, 6.83164465893778212385571895107, 7.44553303887263391257875235573, 8.616008782696301635635136454714, 9.253176209357900988401001777534, 10.07119591672476248371186517129
