| L(s) = 1 | + 3-s + 1.56·5-s − 4.60·7-s + 9-s + 0.677·11-s + 2.42·13-s + 1.56·15-s − 0.541·17-s − 2.86·19-s − 4.60·21-s + 5.52·23-s − 2.56·25-s + 27-s + 4.48·29-s − 6.27·31-s + 0.677·33-s − 7.19·35-s − 4.48·37-s + 2.42·39-s − 2.64·41-s − 10.5·43-s + 1.56·45-s + 0.130·47-s + 14.1·49-s − 0.541·51-s − 1.86·53-s + 1.05·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.698·5-s − 1.74·7-s + 0.333·9-s + 0.204·11-s + 0.671·13-s + 0.403·15-s − 0.131·17-s − 0.657·19-s − 1.00·21-s + 1.15·23-s − 0.512·25-s + 0.192·27-s + 0.833·29-s − 1.12·31-s + 0.117·33-s − 1.21·35-s − 0.737·37-s + 0.387·39-s − 0.413·41-s − 1.60·43-s + 0.232·45-s + 0.0190·47-s + 2.02·49-s − 0.0758·51-s − 0.256·53-s + 0.142·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 - 1.56T + 5T^{2} \) |
| 7 | \( 1 + 4.60T + 7T^{2} \) |
| 11 | \( 1 - 0.677T + 11T^{2} \) |
| 13 | \( 1 - 2.42T + 13T^{2} \) |
| 17 | \( 1 + 0.541T + 17T^{2} \) |
| 19 | \( 1 + 2.86T + 19T^{2} \) |
| 23 | \( 1 - 5.52T + 23T^{2} \) |
| 29 | \( 1 - 4.48T + 29T^{2} \) |
| 31 | \( 1 + 6.27T + 31T^{2} \) |
| 37 | \( 1 + 4.48T + 37T^{2} \) |
| 41 | \( 1 + 2.64T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 - 0.130T + 47T^{2} \) |
| 53 | \( 1 + 1.86T + 53T^{2} \) |
| 59 | \( 1 - 1.40T + 59T^{2} \) |
| 61 | \( 1 + 4.55T + 61T^{2} \) |
| 67 | \( 1 - 2.39T + 67T^{2} \) |
| 71 | \( 1 + 9.71T + 71T^{2} \) |
| 73 | \( 1 - 0.237T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 - 0.539T + 83T^{2} \) |
| 89 | \( 1 - 5.24T + 89T^{2} \) |
| 97 | \( 1 + 18.9T + 97T^{2} \) |
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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
−7.30089963252152765985280532570, −6.73335898540447186796601408841, −6.24337141623729106743125142572, −5.56821876110738625681869670697, −4.59145075432539138371120361289, −3.58704657495712227534352906135, −3.22336427473509524651197851604, −2.32663144974998468172919944471, −1.39093313080999075357337994497, 0,
1.39093313080999075357337994497, 2.32663144974998468172919944471, 3.22336427473509524651197851604, 3.58704657495712227534352906135, 4.59145075432539138371120361289, 5.56821876110738625681869670697, 6.24337141623729106743125142572, 6.73335898540447186796601408841, 7.30089963252152765985280532570
