| L(s) = 1 | − 2.79·3-s − 5-s + 4.79·7-s + 4.79·9-s − 3.79·11-s − 13-s + 2.79·15-s + 17-s − 0.208·19-s − 13.3·21-s + 1.58·23-s − 4·25-s − 4.99·27-s + 9·29-s − 8.58·31-s + 10.5·33-s − 4.79·35-s − 8.58·37-s + 2.79·39-s − 9·43-s − 4.79·45-s + 5.58·47-s + 15.9·49-s − 2.79·51-s − 3.20·53-s + 3.79·55-s + 0.582·57-s + ⋯ |
| L(s) = 1 | − 1.61·3-s − 0.447·5-s + 1.81·7-s + 1.59·9-s − 1.14·11-s − 0.277·13-s + 0.720·15-s + 0.242·17-s − 0.0478·19-s − 2.91·21-s + 0.329·23-s − 0.800·25-s − 0.962·27-s + 1.67·29-s − 1.54·31-s + 1.84·33-s − 0.809·35-s − 1.41·37-s + 0.446·39-s − 1.37·43-s − 0.714·45-s + 0.814·47-s + 2.27·49-s − 0.390·51-s − 0.440·53-s + 0.511·55-s + 0.0771·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 2.79T + 3T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 - 4.79T + 7T^{2} \) |
| 11 | \( 1 + 3.79T + 11T^{2} \) |
| 19 | \( 1 + 0.208T + 19T^{2} \) |
| 23 | \( 1 - 1.58T + 23T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 + 8.58T + 31T^{2} \) |
| 37 | \( 1 + 8.58T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 9T + 43T^{2} \) |
| 47 | \( 1 - 5.58T + 47T^{2} \) |
| 53 | \( 1 + 3.20T + 53T^{2} \) |
| 59 | \( 1 - 0.582T + 59T^{2} \) |
| 61 | \( 1 - 7.20T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + 0.582T + 73T^{2} \) |
| 79 | \( 1 + 0.582T + 79T^{2} \) |
| 83 | \( 1 - 4.58T + 83T^{2} \) |
| 89 | \( 1 - 6.95T + 89T^{2} \) |
| 97 | \( 1 + 7.95T + 97T^{2} \) |
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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
−8.054991676977948232263692444714, −7.44084809471158234839169015823, −6.73743193133315918976026983512, −5.65164537783350556334833270245, −5.10281948304524313975921328591, −4.79801781549219476153015376459, −3.76230793629053182483183411962, −2.25749410450786989788373293403, −1.22304008262650879874074726100, 0,
1.22304008262650879874074726100, 2.25749410450786989788373293403, 3.76230793629053182483183411962, 4.79801781549219476153015376459, 5.10281948304524313975921328591, 5.65164537783350556334833270245, 6.73743193133315918976026983512, 7.44084809471158234839169015823, 8.054991676977948232263692444714
