| L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s − 7-s − 8-s + 9-s + 3·11-s − 2·12-s + 13-s + 14-s + 16-s − 3·17-s − 18-s + 2·19-s + 2·21-s − 3·22-s − 3·23-s + 2·24-s − 5·25-s − 26-s + 4·27-s − 28-s − 4·31-s − 32-s − 6·33-s + 3·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.904·11-s − 0.577·12-s + 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s − 0.235·18-s + 0.458·19-s + 0.436·21-s − 0.639·22-s − 0.625·23-s + 0.408·24-s − 25-s − 0.196·26-s + 0.769·27-s − 0.188·28-s − 0.718·31-s − 0.176·32-s − 1.04·33-s + 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30446 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30446 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 1171 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−15.79128749568019, −14.92386397925478, −14.34139632278437, −13.89092390298417, −13.03636561267707, −12.63162839590496, −11.98754782603600, −11.48063124282411, −11.22062596121268, −10.66578338583999, −9.945204007910299, −9.535059761517382, −8.985476420329441, −8.367768176773600, −7.663799986440690, −7.068985478271498, −6.442938370324703, −6.040039806048648, −5.599169078416441, −4.723377322167726, −4.048709703692319, −3.377601852788726, −2.432979090708622, −1.632003573610394, −0.7959867883163105, 0,
0.7959867883163105, 1.632003573610394, 2.432979090708622, 3.377601852788726, 4.048709703692319, 4.723377322167726, 5.599169078416441, 6.040039806048648, 6.442938370324703, 7.068985478271498, 7.663799986440690, 8.367768176773600, 8.985476420329441, 9.535059761517382, 9.945204007910299, 10.66578338583999, 11.22062596121268, 11.48063124282411, 11.98754782603600, 12.63162839590496, 13.03636561267707, 13.89092390298417, 14.34139632278437, 14.92386397925478, 15.79128749568019
