| L(s) = 1 | + 1.96e4·3-s + 4.64e6·5-s + 8.05e7·7-s + 3.87e8·9-s + 5.09e8·11-s + 1.25e10·13-s + 9.13e10·15-s − 1.17e11·17-s − 1.54e12·19-s + 1.58e12·21-s + 1.34e13·23-s + 2.46e12·25-s + 7.62e12·27-s + 2.32e13·29-s + 2.67e14·31-s + 1.00e13·33-s + 3.74e14·35-s + 1.13e15·37-s + 2.46e14·39-s + 3.27e15·41-s − 2.00e15·43-s + 1.79e15·45-s − 8.39e15·47-s − 4.90e15·49-s − 2.31e15·51-s − 1.22e16·53-s + 2.36e15·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.06·5-s + 0.754·7-s + 0.333·9-s + 0.0652·11-s + 0.327·13-s + 0.613·15-s − 0.240·17-s − 1.09·19-s + 0.435·21-s + 1.55·23-s + 0.129·25-s + 0.192·27-s + 0.297·29-s + 1.81·31-s + 0.0376·33-s + 0.802·35-s + 1.43·37-s + 0.188·39-s + 1.56·41-s − 0.607·43-s + 0.354·45-s − 1.09·47-s − 0.430·49-s − 0.138·51-s − 0.511·53-s + 0.0692·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(\approx\) |
\(3.297486910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.297486910\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.96e4T \) |
| good | 5 | \( 1 - 4.64e6T + 1.90e13T^{2} \) |
| 7 | \( 1 - 8.05e7T + 1.13e16T^{2} \) |
| 11 | \( 1 - 5.09e8T + 6.11e19T^{2} \) |
| 13 | \( 1 - 1.25e10T + 1.46e21T^{2} \) |
| 17 | \( 1 + 1.17e11T + 2.39e23T^{2} \) |
| 19 | \( 1 + 1.54e12T + 1.97e24T^{2} \) |
| 23 | \( 1 - 1.34e13T + 7.46e25T^{2} \) |
| 29 | \( 1 - 2.32e13T + 6.10e27T^{2} \) |
| 31 | \( 1 - 2.67e14T + 2.16e28T^{2} \) |
| 37 | \( 1 - 1.13e15T + 6.24e29T^{2} \) |
| 41 | \( 1 - 3.27e15T + 4.39e30T^{2} \) |
| 43 | \( 1 + 2.00e15T + 1.08e31T^{2} \) |
| 47 | \( 1 + 8.39e15T + 5.88e31T^{2} \) |
| 53 | \( 1 + 1.22e16T + 5.77e32T^{2} \) |
| 59 | \( 1 + 7.92e16T + 4.42e33T^{2} \) |
| 61 | \( 1 + 8.70e16T + 8.34e33T^{2} \) |
| 67 | \( 1 - 2.14e17T + 4.95e34T^{2} \) |
| 71 | \( 1 - 1.28e17T + 1.49e35T^{2} \) |
| 73 | \( 1 - 1.31e17T + 2.53e35T^{2} \) |
| 79 | \( 1 - 1.64e18T + 1.13e36T^{2} \) |
| 83 | \( 1 + 1.82e18T + 2.90e36T^{2} \) |
| 89 | \( 1 + 5.54e18T + 1.09e37T^{2} \) |
| 97 | \( 1 + 7.91e18T + 5.60e37T^{2} \) |
| 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
−15.19426535579432410956147527527, −14.05953326410742733158910934155, −12.96027560437710064656444782943, −11.00948938241342596174105006787, −9.531605013729529301299697503098, −8.214169520317615543952988580251, −6.36436808471956355719556686558, −4.64067979460453200437395451069, −2.61017168555779300847244324211, −1.30847505637927072796519506378,
1.30847505637927072796519506378, 2.61017168555779300847244324211, 4.64067979460453200437395451069, 6.36436808471956355719556686558, 8.214169520317615543952988580251, 9.531605013729529301299697503098, 11.00948938241342596174105006787, 12.96027560437710064656444782943, 14.05953326410742733158910934155, 15.19426535579432410956147527527
