L(s) = 1 | − 0.445·2-s + 2·3-s − 0.801·4-s + 1.24·5-s − 0.890·6-s − 0.445·7-s + 0.801·8-s + 3·9-s − 0.554·10-s − 1.60·12-s − 1.80·13-s + 0.198·14-s + 2.49·15-s + 0.445·16-s + 1.24·17-s − 1.33·18-s − 1.80·19-s − 20-s − 0.890·21-s + 1.60·24-s + 0.554·25-s + 0.801·26-s + 4·27-s + 0.356·28-s − 1.10·30-s − 1.80·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.445·2-s + 2·3-s − 0.801·4-s + 1.24·5-s − 0.890·6-s − 0.445·7-s + 0.801·8-s + 3·9-s − 0.554·10-s − 1.60·12-s − 1.80·13-s + 0.198·14-s + 2.49·15-s + 0.445·16-s + 1.24·17-s − 1.33·18-s − 1.80·19-s − 20-s − 0.890·21-s + 1.60·24-s + 0.554·25-s + 0.801·26-s + 4·27-s + 0.356·28-s − 1.10·30-s − 1.80·31-s − 32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1511 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1511 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.643159580\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.643159580\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 1511 | \( 1 - T \) |
good | 2 | \( 1 + 0.445T + T^{2} \) |
| 3 | \( 1 - 2T + T^{2} \) |
| 5 | \( 1 - 1.24T + T^{2} \) |
| 7 | \( 1 + 0.445T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 1.80T + T^{2} \) |
| 17 | \( 1 - 1.24T + T^{2} \) |
| 19 | \( 1 + 1.80T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + 1.80T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.24T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 0.445T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.519190243123227344640422575381, −9.070367604161419301073657181561, −8.247678810487332628125654678990, −7.56230615817465594115285159840, −6.78793445823414352104747201546, −5.40429109434171825059077269805, −4.44951256295530089453477999415, −3.51020727849849338733585299427, −2.43925821865873943056611733815, −1.73233019483454805620309383781,
1.73233019483454805620309383781, 2.43925821865873943056611733815, 3.51020727849849338733585299427, 4.44951256295530089453477999415, 5.40429109434171825059077269805, 6.78793445823414352104747201546, 7.56230615817465594115285159840, 8.247678810487332628125654678990, 9.070367604161419301073657181561, 9.519190243123227344640422575381