| L(s) = 1 | + 3-s + 5-s + 7-s + 9-s + 5.46·11-s − 3.46·13-s + 15-s + 2·17-s + 5.46·19-s + 21-s + 6.92·23-s + 25-s + 27-s + 2·29-s + 5.46·31-s + 5.46·33-s + 35-s − 2·37-s − 3.46·39-s − 4.92·41-s + 4·43-s + 45-s − 10.9·47-s + 49-s + 2·51-s + 0.535·53-s + 5.46·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s + 0.333·9-s + 1.64·11-s − 0.960·13-s + 0.258·15-s + 0.485·17-s + 1.25·19-s + 0.218·21-s + 1.44·23-s + 0.200·25-s + 0.192·27-s + 0.371·29-s + 0.981·31-s + 0.951·33-s + 0.169·35-s − 0.328·37-s − 0.554·39-s − 0.769·41-s + 0.609·43-s + 0.149·45-s − 1.59·47-s + 0.142·49-s + 0.280·51-s + 0.0736·53-s + 0.736·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.598995273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.598995273\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 5.46T + 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 5.46T + 19T^{2} \) |
| 23 | \( 1 - 6.92T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 5.46T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 4.92T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 - 0.535T + 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 - 4.92T + 61T^{2} \) |
| 67 | \( 1 + 6.92T + 67T^{2} \) |
| 71 | \( 1 + 16.3T + 71T^{2} \) |
| 73 | \( 1 - 0.535T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 - 19.4T + 97T^{2} \) |
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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
−7.960477218782935313877047061817, −7.24670320467552340285557954342, −6.73795803524483661540150029468, −5.91465731724574741411148777371, −4.96648975671923527152762028974, −4.50755989249754518876189545895, −3.38207727875111326199671576545, −2.88375344198569343885894375532, −1.70232720376224871737973823393, −1.05660785103639911918143859502,
1.05660785103639911918143859502, 1.70232720376224871737973823393, 2.88375344198569343885894375532, 3.38207727875111326199671576545, 4.50755989249754518876189545895, 4.96648975671923527152762028974, 5.91465731724574741411148777371, 6.73795803524483661540150029468, 7.24670320467552340285557954342, 7.960477218782935313877047061817
