L(s) = 1 | − 2-s + 2·3-s + 4-s + 2·5-s − 2·6-s − 7-s − 8-s + 9-s − 2·10-s + 11-s + 2·12-s − 4·13-s + 14-s + 4·15-s + 16-s − 18-s + 4·19-s + 2·20-s − 2·21-s − 22-s + 4·23-s − 2·24-s − 25-s + 4·26-s − 4·27-s − 28-s + 2·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.894·5-s − 0.816·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.301·11-s + 0.577·12-s − 1.10·13-s + 0.267·14-s + 1.03·15-s + 1/4·16-s − 0.235·18-s + 0.917·19-s + 0.447·20-s − 0.436·21-s − 0.213·22-s + 0.834·23-s − 0.408·24-s − 1/5·25-s + 0.784·26-s − 0.769·27-s − 0.188·28-s + 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.219984286\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.219984286\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.12237494463202770168598245408, −11.98890555673220470172663809724, −10.59710461621407547188646045730, −9.423423700744115437208349918023, −9.240853610425596933131644882823, −7.87359639769148641923907597256, −6.89070428269537648461178127009, −5.41745538950503066981292745262, −3.29966281651344024951011938340, −2.06919705445307860507213607024,
2.06919705445307860507213607024, 3.29966281651344024951011938340, 5.41745538950503066981292745262, 6.89070428269537648461178127009, 7.87359639769148641923907597256, 9.240853610425596933131644882823, 9.423423700744115437208349918023, 10.59710461621407547188646045730, 11.98890555673220470172663809724, 13.12237494463202770168598245408