| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 9-s − 11-s − 14-s + 16-s − 18-s − 22-s + 25-s − 28-s + 32-s − 36-s − 2·37-s + 2·43-s − 44-s + 49-s + 50-s − 2·53-s − 56-s + 63-s + 64-s − 72-s − 2·74-s + 77-s + 2·79-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 9-s − 11-s − 14-s + 16-s − 18-s − 22-s + 25-s − 28-s + 32-s − 36-s − 2·37-s + 2·43-s − 44-s + 49-s + 50-s − 2·53-s − 56-s + 63-s + 64-s − 72-s − 2·74-s + 77-s + 2·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.194444370\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.194444370\) |
| \(L(1)\) |
|
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 | \( 1 + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( ( 1 + T )^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( ( 1 - T )^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( ( 1 + T )^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( ( 1 - T )^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−12.22444025991043263408956350708, −11.02761844551828325954706001299, −10.41138502805636840024843453249, −9.125716297240622943033830407001, −7.919956933541530537490995495877, −6.82003577140905773126479937248, −5.87497498597523816414932977436, −4.98255767812676813049156880337, −3.46742102224712460117571879696, −2.58191647953742011947022981592,
2.58191647953742011947022981592, 3.46742102224712460117571879696, 4.98255767812676813049156880337, 5.87497498597523816414932977436, 6.82003577140905773126479937248, 7.919956933541530537490995495877, 9.125716297240622943033830407001, 10.41138502805636840024843453249, 11.02761844551828325954706001299, 12.22444025991043263408956350708
