L(s) = 1 | − 2-s + (0.5 − 0.866i)5-s + (1.22 − 0.707i)7-s + 8-s + (−0.5 + 0.866i)10-s + (1.22 + 0.707i)13-s + (−1.22 + 0.707i)14-s − 16-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.499 − 0.866i)25-s + (−1.22 − 0.707i)26-s + 1.41i·29-s − 31-s + (0.5 + 0.866i)34-s − 1.41i·35-s + ⋯ |
L(s) = 1 | − 2-s + (0.5 − 0.866i)5-s + (1.22 − 0.707i)7-s + 8-s + (−0.5 + 0.866i)10-s + (1.22 + 0.707i)13-s + (−1.22 + 0.707i)14-s − 16-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.499 − 0.866i)25-s + (−1.22 − 0.707i)26-s + 1.41i·29-s − 31-s + (0.5 + 0.866i)34-s − 1.41i·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1395 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 + 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1395 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 + 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7897147596\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7897147596\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + T \) |
good | 2 | \( 1 + T + T^{2} \) |
| 7 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - 1.41iT - T^{2} \) |
| 37 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - 1.41iT - T^{2} \) |
| 97 | \( 1 + 1.41iT - T^{2} \) |
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
−9.453383672375112290007035143948, −8.904608151777477589763611661296, −8.214366423094493053421555796408, −7.60019787726207647839359646279, −6.60307924829216227521976509214, −5.33184633293293062446219288344, −4.67909267475133009075645892590, −3.81826097606353480621690909154, −1.78967642330103387947630853523, −1.18153870271312122993747494423,
1.44370858016785837672939890686, 2.40435815726732939582308112373, 3.77827789311740804134618293433, 4.94782615428288859444366676829, 5.81156700556449407604778750635, 6.70834328450601136444466664554, 7.81322203546979052137637347638, 8.268462949820081962940929204819, 9.000794557508510515623688096825, 9.787920520246646012663422078723