| L(s) = 1 | + 3-s + 5-s + 7-s − 2·9-s + 13-s + 15-s + 2·17-s − 5·19-s + 21-s − 23-s − 4·25-s − 5·27-s + 10·29-s + 8·31-s + 35-s − 2·37-s + 39-s − 7·41-s − 6·43-s − 2·45-s − 3·47-s + 49-s + 2·51-s − 6·53-s − 5·57-s − 12·61-s − 2·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s − 2/3·9-s + 0.277·13-s + 0.258·15-s + 0.485·17-s − 1.14·19-s + 0.218·21-s − 0.208·23-s − 4/5·25-s − 0.962·27-s + 1.85·29-s + 1.43·31-s + 0.169·35-s − 0.328·37-s + 0.160·39-s − 1.09·41-s − 0.914·43-s − 0.298·45-s − 0.437·47-s + 1/7·49-s + 0.280·51-s − 0.824·53-s − 0.662·57-s − 1.53·61-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 311696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.921398783\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.921398783\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−12.60706608480728, −12.17060099298078, −11.75647627259990, −11.38382681872425, −10.67738889700463, −10.39915992858354, −9.853871453454984, −9.525888193449050, −8.811657745710268, −8.427750594232914, −8.188413873301900, −7.796769329129799, −6.992556046281533, −6.492295765813287, −6.140605160913146, −5.643227217285344, −5.018842242069073, −4.541081569228332, −4.062697467041165, −3.315576129126732, −2.897999047330838, −2.454899913573238, −1.681609520548743, −1.381199598387125, −0.3310799849105199,
0.3310799849105199, 1.381199598387125, 1.681609520548743, 2.454899913573238, 2.897999047330838, 3.315576129126732, 4.062697467041165, 4.541081569228332, 5.018842242069073, 5.643227217285344, 6.140605160913146, 6.492295765813287, 6.992556046281533, 7.796769329129799, 8.188413873301900, 8.427750594232914, 8.811657745710268, 9.525888193449050, 9.853871453454984, 10.39915992858354, 10.67738889700463, 11.38382681872425, 11.75647627259990, 12.17060099298078, 12.60706608480728
