L(s) = 1 | + (0.437 + 0.437i)2-s − 0.618i·4-s + (−0.987 − 0.156i)5-s + (−0.642 + 0.642i)7-s + (0.707 − 0.707i)8-s + (−0.363 − 0.5i)10-s + (1.39 + 1.39i)13-s − 0.561·14-s + (−0.0966 + 0.610i)20-s + (1.34 − 1.34i)23-s + (0.951 + 0.309i)25-s + 1.22i·26-s + (0.396 + 0.396i)28-s + (−0.707 − 0.707i)32-s + (0.734 − 0.533i)35-s + ⋯ |
L(s) = 1 | + (0.437 + 0.437i)2-s − 0.618i·4-s + (−0.987 − 0.156i)5-s + (−0.642 + 0.642i)7-s + (0.707 − 0.707i)8-s + (−0.363 − 0.5i)10-s + (1.39 + 1.39i)13-s − 0.561·14-s + (−0.0966 + 0.610i)20-s + (1.34 − 1.34i)23-s + (0.951 + 0.309i)25-s + 1.22i·26-s + (0.396 + 0.396i)28-s + (−0.707 − 0.707i)32-s + (0.734 − 0.533i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.304059417\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.304059417\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.987 + 0.156i)T \) |
| 53 | \( 1 + (-0.707 + 0.707i)T \) |
good | 2 | \( 1 + (-0.437 - 0.437i)T + iT^{2} \) |
| 7 | \( 1 + (0.642 - 0.642i)T - iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (-1.39 - 1.39i)T + iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (-1.34 + 1.34i)T - iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-1.26 + 1.26i)T - iT^{2} \) |
| 41 | \( 1 + 0.907iT - T^{2} \) |
| 43 | \( 1 + (-0.221 - 0.221i)T + iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 - 1.97iT - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + (0.831 - 0.831i)T - iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.221 + 0.221i)T - iT^{2} \) |
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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
−8.950361567851638159523468974729, −8.612248409596782427639781757661, −7.37609064779131334691810985231, −6.70384167375831558933503080206, −6.15814282881366378309211973075, −5.26514031119891564254918885592, −4.31184623655916761633751443989, −3.80268738593218170562265816483, −2.50661792629145290056386901608, −1.04171931491444531368818228823,
1.11453472166033040283013546191, 3.01985041157136213918619628359, 3.29241782299955602921405867327, 4.06663784184373585624504802437, 4.94773699718279036657986722436, 6.02280653992593419708723351926, 7.00622704132543432037759446800, 7.67995670649498011129855584756, 8.217492546346832642245623093330, 9.006423327748631158321514169512