| L(s) = 1 | − 1.79·3-s + 2.79·5-s + 0.208·9-s + 5.58·11-s − 4·13-s − 5·15-s − 17-s − 2·19-s − 7.58·23-s + 2.79·25-s + 5.00·27-s − 4.79·31-s − 10·33-s − 3.58·37-s + 7.16·39-s − 5.79·41-s + 1.79·43-s + 0.582·45-s − 7.58·47-s + 1.79·51-s − 5.20·53-s + 15.5·55-s + 3.58·57-s − 2.41·59-s − 2.20·61-s − 11.1·65-s + 11.9·67-s + ⋯ |
| L(s) = 1 | − 1.03·3-s + 1.24·5-s + 0.0695·9-s + 1.68·11-s − 1.10·13-s − 1.29·15-s − 0.242·17-s − 0.458·19-s − 1.58·23-s + 0.558·25-s + 0.962·27-s − 0.860·31-s − 1.74·33-s − 0.588·37-s + 1.14·39-s − 0.904·41-s + 0.273·43-s + 0.0868·45-s − 1.10·47-s + 0.250·51-s − 0.715·53-s + 2.10·55-s + 0.474·57-s − 0.314·59-s − 0.282·61-s − 1.38·65-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 1.79T + 3T^{2} \) |
| 5 | \( 1 - 2.79T + 5T^{2} \) |
| 11 | \( 1 - 5.58T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 7.58T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 4.79T + 31T^{2} \) |
| 37 | \( 1 + 3.58T + 37T^{2} \) |
| 41 | \( 1 + 5.79T + 41T^{2} \) |
| 43 | \( 1 - 1.79T + 43T^{2} \) |
| 47 | \( 1 + 7.58T + 47T^{2} \) |
| 53 | \( 1 + 5.20T + 53T^{2} \) |
| 59 | \( 1 + 2.41T + 59T^{2} \) |
| 61 | \( 1 + 2.20T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 - 0.417T + 83T^{2} \) |
| 89 | \( 1 + 7.16T + 89T^{2} \) |
| 97 | \( 1 + 5.20T + 97T^{2} \) |
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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
−8.394574882230178462323055196187, −7.22476923422507437284312042975, −6.44732848220904090216194438674, −6.12174373467520294179781685223, −5.32963711216213509309440371141, −4.60575524283424273019311188737, −3.60813392503252034933952291889, −2.25486934919605927145213512251, −1.51587604481610163478791928692, 0,
1.51587604481610163478791928692, 2.25486934919605927145213512251, 3.60813392503252034933952291889, 4.60575524283424273019311188737, 5.32963711216213509309440371141, 6.12174373467520294179781685223, 6.44732848220904090216194438674, 7.22476923422507437284312042975, 8.394574882230178462323055196187
