L(s) = 1 | + 2-s + 4-s − 5-s − 4·7-s + 8-s − 10-s − 4·14-s + 16-s + 4·17-s + 8·19-s − 20-s − 23-s + 25-s − 4·28-s − 2·29-s + 4·31-s + 32-s + 4·34-s + 4·35-s − 2·37-s + 8·38-s − 40-s + 8·41-s − 4·43-s − 46-s + 8·47-s + 9·49-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.51·7-s + 0.353·8-s − 0.316·10-s − 1.06·14-s + 1/4·16-s + 0.970·17-s + 1.83·19-s − 0.223·20-s − 0.208·23-s + 1/5·25-s − 0.755·28-s − 0.371·29-s + 0.718·31-s + 0.176·32-s + 0.685·34-s + 0.676·35-s − 0.328·37-s + 1.29·38-s − 0.158·40-s + 1.24·41-s − 0.609·43-s − 0.147·46-s + 1.16·47-s + 9/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.218657191\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.218657191\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−9.363759421999071551519873219897, −8.169544344054336530160868215918, −7.36482822759136563408482202330, −6.76905828865410232001666142108, −5.83274984474283808765894101542, −5.24735210826333178392507769876, −3.98633328640209808605653607971, −3.38301548447067057978221612054, −2.63248985083715070637125266136, −0.894778098781870952960807550841,
0.894778098781870952960807550841, 2.63248985083715070637125266136, 3.38301548447067057978221612054, 3.98633328640209808605653607971, 5.24735210826333178392507769876, 5.83274984474283808765894101542, 6.76905828865410232001666142108, 7.36482822759136563408482202330, 8.169544344054336530160868215918, 9.363759421999071551519873219897