| L(s) = 1 | + 0.630·3-s + 0.354·5-s − 2.60·9-s + 3.19·11-s − 1.78·13-s + 0.223·15-s + 4.94·17-s − 0.00623·19-s − 23-s − 4.87·25-s − 3.53·27-s + 6.91·29-s − 9.14·31-s + 2.01·33-s − 8.81·37-s − 1.12·39-s + 1.10·41-s + 5.54·43-s − 0.922·45-s − 10.4·47-s + 3.11·51-s − 10.9·53-s + 1.13·55-s − 0.00393·57-s + 12.3·59-s + 1.89·61-s − 0.633·65-s + ⋯ |
| L(s) = 1 | + 0.364·3-s + 0.158·5-s − 0.867·9-s + 0.964·11-s − 0.496·13-s + 0.0576·15-s + 1.19·17-s − 0.00142·19-s − 0.208·23-s − 0.974·25-s − 0.679·27-s + 1.28·29-s − 1.64·31-s + 0.351·33-s − 1.44·37-s − 0.180·39-s + 0.172·41-s + 0.845·43-s − 0.137·45-s − 1.52·47-s + 0.436·51-s − 1.50·53-s + 0.152·55-s − 0.000520·57-s + 1.61·59-s + 0.242·61-s − 0.0786·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 0.630T + 3T^{2} \) |
| 5 | \( 1 - 0.354T + 5T^{2} \) |
| 11 | \( 1 - 3.19T + 11T^{2} \) |
| 13 | \( 1 + 1.78T + 13T^{2} \) |
| 17 | \( 1 - 4.94T + 17T^{2} \) |
| 19 | \( 1 + 0.00623T + 19T^{2} \) |
| 29 | \( 1 - 6.91T + 29T^{2} \) |
| 31 | \( 1 + 9.14T + 31T^{2} \) |
| 37 | \( 1 + 8.81T + 37T^{2} \) |
| 41 | \( 1 - 1.10T + 41T^{2} \) |
| 43 | \( 1 - 5.54T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 - 1.89T + 61T^{2} \) |
| 67 | \( 1 + 4.95T + 67T^{2} \) |
| 71 | \( 1 + 4.45T + 71T^{2} \) |
| 73 | \( 1 + 4.55T + 73T^{2} \) |
| 79 | \( 1 + 7.24T + 79T^{2} \) |
| 83 | \( 1 - 2.10T + 83T^{2} \) |
| 89 | \( 1 + 6.30T + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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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
−7.44402023058313400431869875546, −6.75363636448802151116958828710, −5.92307798827765465794203749227, −5.46817117533280177397037750214, −4.58850620124396346968865882072, −3.64249212937983607397019961708, −3.18067265898022682912816439153, −2.18573412490135766309085496794, −1.36069115584503665607653910031, 0,
1.36069115584503665607653910031, 2.18573412490135766309085496794, 3.18067265898022682912816439153, 3.64249212937983607397019961708, 4.58850620124396346968865882072, 5.46817117533280177397037750214, 5.92307798827765465794203749227, 6.75363636448802151116958828710, 7.44402023058313400431869875546
