| L(s) = 1 | + 2-s + 4-s + 2·5-s − 7-s + 8-s + 2·10-s − 4·11-s + 6·13-s − 14-s + 16-s + 2·17-s + 4·19-s + 2·20-s − 4·22-s − 8·23-s − 25-s + 6·26-s − 28-s + 32-s + 2·34-s − 2·35-s + 10·37-s + 4·38-s + 2·40-s − 6·41-s + 4·43-s − 4·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s + 0.353·8-s + 0.632·10-s − 1.20·11-s + 1.66·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s + 0.917·19-s + 0.447·20-s − 0.852·22-s − 1.66·23-s − 1/5·25-s + 1.17·26-s − 0.188·28-s + 0.176·32-s + 0.342·34-s − 0.338·35-s + 1.64·37-s + 0.648·38-s + 0.316·40-s − 0.937·41-s + 0.609·43-s − 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 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 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.85660574310879, −13.36486917900637, −13.20874606673305, −12.65666221661443, −11.99117662730688, −11.62791543018648, −10.97857124481096, −10.52868905901519, −10.05833465541661, −9.667611048960616, −9.087191667682297, −8.390620844279905, −7.772857249694640, −7.624251666701913, −6.651973552612577, −6.084924110458726, −5.906713655173372, −5.443826945078380, −4.772614363865631, −4.080502087004270, −3.546037221344114, −2.951410618830526, −2.424667105371893, −1.671911632745683, −1.124952159352948, 0,
1.124952159352948, 1.671911632745683, 2.424667105371893, 2.951410618830526, 3.546037221344114, 4.080502087004270, 4.772614363865631, 5.443826945078380, 5.906713655173372, 6.084924110458726, 6.651973552612577, 7.624251666701913, 7.772857249694640, 8.390620844279905, 9.087191667682297, 9.667611048960616, 10.05833465541661, 10.52868905901519, 10.97857124481096, 11.62791543018648, 11.99117662730688, 12.65666221661443, 13.20874606673305, 13.36486917900637, 13.85660574310879
