L(s) = 1 | − 5-s − 3·9-s − 2·13-s − 17-s + 4·19-s + 8·23-s + 25-s − 2·29-s − 8·31-s − 2·37-s − 2·41-s − 4·43-s + 3·45-s − 6·53-s + 4·59-s − 6·61-s + 2·65-s + 4·67-s + 8·71-s − 2·73-s + 9·81-s − 4·83-s + 85-s + 6·89-s − 4·95-s − 18·97-s + 101-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 9-s − 0.554·13-s − 0.242·17-s + 0.917·19-s + 1.66·23-s + 1/5·25-s − 0.371·29-s − 1.43·31-s − 0.328·37-s − 0.312·41-s − 0.609·43-s + 0.447·45-s − 0.824·53-s + 0.520·59-s − 0.768·61-s + 0.248·65-s + 0.488·67-s + 0.949·71-s − 0.234·73-s + 81-s − 0.439·83-s + 0.108·85-s + 0.635·89-s − 0.410·95-s − 1.82·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 266560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 266560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−12.86690016337600, −12.64858083728281, −12.04568601681158, −11.57207769769916, −11.18377445990406, −10.92995610739169, −10.32393880627128, −9.640317018951648, −9.346913076295485, −8.769311403364211, −8.508716858742072, −7.782026227048134, −7.444417996521567, −6.952047390146736, −6.513990901234153, −5.772718151776961, −5.336466712099703, −4.972168381417782, −4.447422586049827, −3.547248669192178, −3.355833906951338, −2.774677605925875, −2.140791116305849, −1.444880779501911, −0.6555124544871757, 0,
0.6555124544871757, 1.444880779501911, 2.140791116305849, 2.774677605925875, 3.355833906951338, 3.547248669192178, 4.447422586049827, 4.972168381417782, 5.336466712099703, 5.772718151776961, 6.513990901234153, 6.952047390146736, 7.444417996521567, 7.782026227048134, 8.508716858742072, 8.769311403364211, 9.346913076295485, 9.640317018951648, 10.32393880627128, 10.92995610739169, 11.18377445990406, 11.57207769769916, 12.04568601681158, 12.64858083728281, 12.86690016337600