L(s) = 1 | + (0.142 − 0.989i)2-s + (−0.959 − 0.281i)4-s + (−0.114 − 0.0855i)5-s + (−0.415 + 0.909i)8-s + (0.281 + 0.959i)9-s + (−0.100 + 0.100i)10-s + (1.17 − 0.254i)13-s + (0.841 + 0.540i)16-s + (0.797 − 1.74i)17-s + (0.989 − 0.142i)18-s + (0.0855 + 0.114i)20-s + (−0.275 − 0.939i)25-s + (−0.0855 − 1.19i)26-s + (1.03 − 0.304i)29-s + (0.654 − 0.755i)32-s + ⋯ |
L(s) = 1 | + (0.142 − 0.989i)2-s + (−0.959 − 0.281i)4-s + (−0.114 − 0.0855i)5-s + (−0.415 + 0.909i)8-s + (0.281 + 0.959i)9-s + (−0.100 + 0.100i)10-s + (1.17 − 0.254i)13-s + (0.841 + 0.540i)16-s + (0.797 − 1.74i)17-s + (0.989 − 0.142i)18-s + (0.0855 + 0.114i)20-s + (−0.275 − 0.939i)25-s + (−0.0855 − 1.19i)26-s + (1.03 − 0.304i)29-s + (0.654 − 0.755i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.120 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.120 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.103796786\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.103796786\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.142 + 0.989i)T \) |
| 353 | \( 1 + (-0.415 - 0.909i)T \) |
good | 3 | \( 1 + (-0.281 - 0.959i)T^{2} \) |
| 5 | \( 1 + (0.114 + 0.0855i)T + (0.281 + 0.959i)T^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 13 | \( 1 + (-1.17 + 0.254i)T + (0.909 - 0.415i)T^{2} \) |
| 17 | \( 1 + (-0.797 + 1.74i)T + (-0.654 - 0.755i)T^{2} \) |
| 19 | \( 1 + (0.415 - 0.909i)T^{2} \) |
| 23 | \( 1 + (-0.142 - 0.989i)T^{2} \) |
| 29 | \( 1 + (-1.03 + 0.304i)T + (0.841 - 0.540i)T^{2} \) |
| 31 | \( 1 + (-0.989 - 0.142i)T^{2} \) |
| 37 | \( 1 + (1.75 - 0.654i)T + (0.755 - 0.654i)T^{2} \) |
| 41 | \( 1 + (-0.627 + 0.544i)T + (0.142 - 0.989i)T^{2} \) |
| 43 | \( 1 + (-0.142 + 0.989i)T^{2} \) |
| 47 | \( 1 + (0.841 + 0.540i)T^{2} \) |
| 53 | \( 1 + (0.574 - 0.767i)T + (-0.281 - 0.959i)T^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + (-0.909 - 0.584i)T + (0.415 + 0.909i)T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 + (0.540 - 0.841i)T^{2} \) |
| 73 | \( 1 + (0.841 + 0.540i)T^{2} \) |
| 79 | \( 1 + (-0.281 + 0.959i)T^{2} \) |
| 83 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 89 | \( 1 + (-0.133 - 0.0498i)T + (0.755 + 0.654i)T^{2} \) |
| 97 | \( 1 + (0.118 + 0.258i)T + (-0.654 + 0.755i)T^{2} \) |
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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
−9.805331385848206624412848654458, −8.822966418200132030984583164697, −8.176971545023338692324983133462, −7.30067141248221286944611929312, −6.03382365594003728186973454163, −5.13302572722162012675313731402, −4.43709608099304336107698480216, −3.34009770464615006662614845210, −2.44146042055587003077105448834, −1.11736088699219753603585621097,
1.36508547971127231589829636037, 3.57196898432784001882833483327, 3.76924499684811115470998542328, 5.09619135948211884747303137080, 6.05294772795580123038782671932, 6.53201193904005897621764185543, 7.43213005033141331172650448231, 8.391085913821902321550477857723, 8.821205146838205352017775211111, 9.791213446421519444690063534476