L(s) = 1 | − 2-s + 4-s − 5-s − 3·7-s − 8-s − 3·9-s + 10-s + 11-s − 2·13-s + 3·14-s + 16-s − 17-s + 3·18-s − 7·19-s − 20-s − 22-s + 9·23-s − 4·25-s + 2·26-s − 3·28-s − 6·29-s − 10·31-s − 32-s + 34-s + 3·35-s − 3·36-s + 4·37-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.13·7-s − 0.353·8-s − 9-s + 0.316·10-s + 0.301·11-s − 0.554·13-s + 0.801·14-s + 1/4·16-s − 0.242·17-s + 0.707·18-s − 1.60·19-s − 0.223·20-s − 0.213·22-s + 1.87·23-s − 4/5·25-s + 0.392·26-s − 0.566·28-s − 1.11·29-s − 1.79·31-s − 0.176·32-s + 0.171·34-s + 0.507·35-s − 1/2·36-s + 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 127 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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
−11.30744232254828095648588604529, −10.70024523918434590003033137629, −9.339551086476144119275448247139, −8.918319006697253450920486424331, −7.62645923990918495315373080588, −6.69000360425299706559888754932, −5.63597740391869263098869003051, −3.84272287311560698426907861601, −2.54210904410735093209815951472, 0,
2.54210904410735093209815951472, 3.84272287311560698426907861601, 5.63597740391869263098869003051, 6.69000360425299706559888754932, 7.62645923990918495315373080588, 8.918319006697253450920486424331, 9.339551086476144119275448247139, 10.70024523918434590003033137629, 11.30744232254828095648588604529