| L(s) = 1 | − 0.923·3-s + 1.20·5-s + 1.96·7-s − 2.14·9-s − 0.336·11-s − 7.20·13-s − 1.11·15-s + 5.33·17-s − 5.23·19-s − 1.81·21-s − 3.74·23-s − 3.54·25-s + 4.75·27-s + 6.55·29-s − 4.05·31-s + 0.310·33-s + 2.36·35-s + 10.0·37-s + 6.65·39-s − 1.65·41-s + 9.79·43-s − 2.58·45-s + 6.95·47-s − 3.15·49-s − 4.93·51-s + 9.58·53-s − 0.405·55-s + ⋯ |
| L(s) = 1 | − 0.533·3-s + 0.539·5-s + 0.740·7-s − 0.715·9-s − 0.101·11-s − 1.99·13-s − 0.287·15-s + 1.29·17-s − 1.20·19-s − 0.395·21-s − 0.781·23-s − 0.709·25-s + 0.914·27-s + 1.21·29-s − 0.728·31-s + 0.0540·33-s + 0.399·35-s + 1.65·37-s + 1.06·39-s − 0.258·41-s + 1.49·43-s − 0.385·45-s + 1.01·47-s − 0.451·49-s − 0.690·51-s + 1.31·53-s − 0.0546·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.397133384\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.397133384\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 + 0.923T + 3T^{2} \) |
| 5 | \( 1 - 1.20T + 5T^{2} \) |
| 7 | \( 1 - 1.96T + 7T^{2} \) |
| 11 | \( 1 + 0.336T + 11T^{2} \) |
| 13 | \( 1 + 7.20T + 13T^{2} \) |
| 17 | \( 1 - 5.33T + 17T^{2} \) |
| 19 | \( 1 + 5.23T + 19T^{2} \) |
| 23 | \( 1 + 3.74T + 23T^{2} \) |
| 29 | \( 1 - 6.55T + 29T^{2} \) |
| 31 | \( 1 + 4.05T + 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 + 1.65T + 41T^{2} \) |
| 43 | \( 1 - 9.79T + 43T^{2} \) |
| 47 | \( 1 - 6.95T + 47T^{2} \) |
| 53 | \( 1 - 9.58T + 53T^{2} \) |
| 59 | \( 1 - 2.36T + 59T^{2} \) |
| 61 | \( 1 - 4.90T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 - 4.32T + 71T^{2} \) |
| 73 | \( 1 - 6.12T + 73T^{2} \) |
| 79 | \( 1 + 14.0T + 79T^{2} \) |
| 83 | \( 1 - 7.96T + 83T^{2} \) |
| 89 | \( 1 - 9.79T + 89T^{2} \) |
| 97 | \( 1 + 9.23T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.259104393104534037200027577866, −7.81149290807956143415953010587, −6.97781347484777425009326840940, −6.02062695054456336950513111303, −5.51824784680871255953545897136, −4.85074129843297095272180987285, −4.03742047008813707197709556267, −2.63689933356932893160148027597, −2.14497802952110549366611283290, −0.67318842915821124246011266655,
0.67318842915821124246011266655, 2.14497802952110549366611283290, 2.63689933356932893160148027597, 4.03742047008813707197709556267, 4.85074129843297095272180987285, 5.51824784680871255953545897136, 6.02062695054456336950513111303, 6.97781347484777425009326840940, 7.81149290807956143415953010587, 8.259104393104534037200027577866
