L(s) = 1 | + 3-s − 4·5-s + 7-s + 9-s + 2·11-s − 4·13-s − 4·15-s + 4·17-s + 19-s + 21-s − 6·23-s + 11·25-s + 27-s − 2·29-s + 4·31-s + 2·33-s − 4·35-s + 2·37-s − 4·39-s − 2·41-s − 8·43-s − 4·45-s + 4·47-s + 49-s + 4·51-s − 10·53-s − 8·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.78·5-s + 0.377·7-s + 1/3·9-s + 0.603·11-s − 1.10·13-s − 1.03·15-s + 0.970·17-s + 0.229·19-s + 0.218·21-s − 1.25·23-s + 11/5·25-s + 0.192·27-s − 0.371·29-s + 0.718·31-s + 0.348·33-s − 0.676·35-s + 0.328·37-s − 0.640·39-s − 0.312·41-s − 1.21·43-s − 0.596·45-s + 0.583·47-s + 1/7·49-s + 0.560·51-s − 1.37·53-s − 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−8.202014148017668111109405230205, −7.55193439322751323690271804805, −7.24625637331685549555478863003, −6.11147651573255497645585639759, −4.90470188476135481545855376546, −4.30195768525910966533150394971, −3.56519765938617052854028891356, −2.80330358213221290594941201662, −1.43583775049627081915768854901, 0,
1.43583775049627081915768854901, 2.80330358213221290594941201662, 3.56519765938617052854028891356, 4.30195768525910966533150394971, 4.90470188476135481545855376546, 6.11147651573255497645585639759, 7.24625637331685549555478863003, 7.55193439322751323690271804805, 8.202014148017668111109405230205