| L(s) = 1 | + 2·3-s + 5-s − 7-s + 9-s − 11-s − 4·13-s + 2·15-s + 4·19-s − 2·21-s − 6·23-s + 25-s − 4·27-s − 6·29-s + 4·31-s − 2·33-s − 35-s + 2·37-s − 8·39-s − 6·41-s + 4·43-s + 45-s − 6·47-s + 49-s − 6·53-s − 55-s + 8·57-s − 10·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.10·13-s + 0.516·15-s + 0.917·19-s − 0.436·21-s − 1.25·23-s + 1/5·25-s − 0.769·27-s − 1.11·29-s + 0.718·31-s − 0.348·33-s − 0.169·35-s + 0.328·37-s − 1.28·39-s − 0.937·41-s + 0.609·43-s + 0.149·45-s − 0.875·47-s + 1/7·49-s − 0.824·53-s − 0.134·55-s + 1.05·57-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.82342711556442167952183913727, −7.20911502235371928782314700522, −6.30646359554401990964179147553, −5.56537613833728450166754495338, −4.79671730557479680048447735345, −3.83283897965637988592917753074, −3.06212144706437026169746579522, −2.45312318073906820960660641445, −1.64133394497705981625654684434, 0,
1.64133394497705981625654684434, 2.45312318073906820960660641445, 3.06212144706437026169746579522, 3.83283897965637988592917753074, 4.79671730557479680048447735345, 5.56537613833728450166754495338, 6.30646359554401990964179147553, 7.20911502235371928782314700522, 7.82342711556442167952183913727
