| L(s) = 1 | + 2.73·3-s − 3.07·5-s + 4.47·9-s − 0.693·11-s + 1.77·13-s − 8.40·15-s + 1.86·17-s + 2.98·19-s − 23-s + 4.45·25-s + 4.04·27-s + 2.84·29-s − 1.48·31-s − 1.89·33-s + 1.47·37-s + 4.85·39-s − 10.3·41-s + 0.231·43-s − 13.7·45-s + 7.58·47-s + 5.10·51-s + 4.64·53-s + 2.13·55-s + 8.17·57-s + 8.95·59-s − 2.24·61-s − 5.45·65-s + ⋯ |
| L(s) = 1 | + 1.57·3-s − 1.37·5-s + 1.49·9-s − 0.209·11-s + 0.492·13-s − 2.17·15-s + 0.452·17-s + 0.685·19-s − 0.208·23-s + 0.890·25-s + 0.777·27-s + 0.527·29-s − 0.267·31-s − 0.329·33-s + 0.242·37-s + 0.777·39-s − 1.61·41-s + 0.0353·43-s − 2.05·45-s + 1.10·47-s + 0.714·51-s + 0.638·53-s + 0.287·55-s + 1.08·57-s + 1.16·59-s − 0.287·61-s − 0.676·65-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
| \(L(1)\) |
\(\approx\) |
\(2.936268348\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.936268348\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.73T + 3T^{2} \) |
| 5 | \( 1 + 3.07T + 5T^{2} \) |
| 11 | \( 1 + 0.693T + 11T^{2} \) |
| 13 | \( 1 - 1.77T + 13T^{2} \) |
| 17 | \( 1 - 1.86T + 17T^{2} \) |
| 19 | \( 1 - 2.98T + 19T^{2} \) |
| 29 | \( 1 - 2.84T + 29T^{2} \) |
| 31 | \( 1 + 1.48T + 31T^{2} \) |
| 37 | \( 1 - 1.47T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 0.231T + 43T^{2} \) |
| 47 | \( 1 - 7.58T + 47T^{2} \) |
| 53 | \( 1 - 4.64T + 53T^{2} \) |
| 59 | \( 1 - 8.95T + 59T^{2} \) |
| 61 | \( 1 + 2.24T + 61T^{2} \) |
| 67 | \( 1 - 2.85T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 - 3.50T + 73T^{2} \) |
| 79 | \( 1 - 7.42T + 79T^{2} \) |
| 83 | \( 1 - 5.90T + 83T^{2} \) |
| 89 | \( 1 - 17.1T + 89T^{2} \) |
| 97 | \( 1 - 4.64T + 97T^{2} \) |
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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
−7.81060804220755757141410476485, −7.38654776928616026805709137263, −6.69008832218899440438490937080, −5.59685218819249425907466214457, −4.69672715274630019156428445749, −3.89508864783298727850518470825, −3.48917168962978264023563681408, −2.85991399205637437309339891136, −1.91125090111544229574886332345, −0.76831554304239583647898207331,
0.76831554304239583647898207331, 1.91125090111544229574886332345, 2.85991399205637437309339891136, 3.48917168962978264023563681408, 3.89508864783298727850518470825, 4.69672715274630019156428445749, 5.59685218819249425907466214457, 6.69008832218899440438490937080, 7.38654776928616026805709137263, 7.81060804220755757141410476485
