| L(s) = 1 | + 3-s − 5-s + 9-s − 2·11-s + 6·13-s − 15-s + 6·17-s − 2·19-s + 25-s + 27-s + 2·29-s − 2·33-s − 37-s + 6·39-s + 2·41-s + 8·43-s − 45-s + 6·47-s − 7·49-s + 6·51-s + 2·53-s + 2·55-s − 2·57-s − 8·59-s − 8·61-s − 6·65-s − 4·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.603·11-s + 1.66·13-s − 0.258·15-s + 1.45·17-s − 0.458·19-s + 1/5·25-s + 0.192·27-s + 0.371·29-s − 0.348·33-s − 0.164·37-s + 0.960·39-s + 0.312·41-s + 1.21·43-s − 0.149·45-s + 0.875·47-s − 49-s + 0.840·51-s + 0.274·53-s + 0.269·55-s − 0.264·57-s − 1.04·59-s − 1.02·61-s − 0.744·65-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.658694822\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.658694822\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−7.72456858959374742096421244825, −7.35945566441120010272212861903, −6.24689054768640776108621623734, −5.85145308387066181355913750940, −4.87396409035489516308035625781, −4.09275293104405961413694716969, −3.41555781057969416984491118939, −2.83787059934029233183128462455, −1.69893067979067632476540382737, −0.803148560832975155921498702867,
0.803148560832975155921498702867, 1.69893067979067632476540382737, 2.83787059934029233183128462455, 3.41555781057969416984491118939, 4.09275293104405961413694716969, 4.87396409035489516308035625781, 5.85145308387066181355913750940, 6.24689054768640776108621623734, 7.35945566441120010272212861903, 7.72456858959374742096421244825
