| L(s) = 1 | + 2-s + 4-s + 1.53·5-s + 5.12·7-s + 8-s + 1.53·10-s − 4.12·11-s + 2.46·13-s + 5.12·14-s + 16-s − 4.78·17-s − 1.65·19-s + 1.53·20-s − 4.12·22-s + 5.18·23-s − 2.65·25-s + 2.46·26-s + 5.12·28-s + 0.871·29-s + 5.91·31-s + 32-s − 4.78·34-s + 7.84·35-s + 8.24·37-s − 1.65·38-s + 1.53·40-s + 9.17·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.684·5-s + 1.93·7-s + 0.353·8-s + 0.483·10-s − 1.24·11-s + 0.685·13-s + 1.37·14-s + 0.250·16-s − 1.16·17-s − 0.380·19-s + 0.342·20-s − 0.880·22-s + 1.08·23-s − 0.531·25-s + 0.484·26-s + 0.969·28-s + 0.161·29-s + 1.06·31-s + 0.176·32-s − 0.820·34-s + 1.32·35-s + 1.35·37-s − 0.269·38-s + 0.241·40-s + 1.43·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.323274528\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.323274528\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 223 | \( 1 - T \) |
| good | 5 | \( 1 - 1.53T + 5T^{2} \) |
| 7 | \( 1 - 5.12T + 7T^{2} \) |
| 11 | \( 1 + 4.12T + 11T^{2} \) |
| 13 | \( 1 - 2.46T + 13T^{2} \) |
| 17 | \( 1 + 4.78T + 17T^{2} \) |
| 19 | \( 1 + 1.65T + 19T^{2} \) |
| 23 | \( 1 - 5.18T + 23T^{2} \) |
| 29 | \( 1 - 0.871T + 29T^{2} \) |
| 31 | \( 1 - 5.91T + 31T^{2} \) |
| 37 | \( 1 - 8.24T + 37T^{2} \) |
| 41 | \( 1 - 9.17T + 41T^{2} \) |
| 43 | \( 1 + 6.91T + 43T^{2} \) |
| 47 | \( 1 - 6.72T + 47T^{2} \) |
| 53 | \( 1 + 6.71T + 53T^{2} \) |
| 59 | \( 1 + 1.59T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 - 8.59T + 67T^{2} \) |
| 71 | \( 1 - 0.743T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 + 5.70T + 79T^{2} \) |
| 83 | \( 1 + 7.79T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + 12.1T + 97T^{2} \) |
| 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
−8.302035302754145851064664913816, −7.78438964042295986914849197475, −6.92429505841446988530751247903, −5.98685420476351020406840193754, −5.40382253388563416305565167426, −4.66912478080721850522319444198, −4.20257617562639338707166730792, −2.71672351326373620610274619553, −2.16747218917727764414918443977, −1.17033166431240047115197318848,
1.17033166431240047115197318848, 2.16747218917727764414918443977, 2.71672351326373620610274619553, 4.20257617562639338707166730792, 4.66912478080721850522319444198, 5.40382253388563416305565167426, 5.98685420476351020406840193754, 6.92429505841446988530751247903, 7.78438964042295986914849197475, 8.302035302754145851064664913816
