L(s) = 1 | − 0.906·2-s + 3.21·3-s − 1.17·4-s − 2.91·6-s + 2.59·7-s + 2.88·8-s + 7.35·9-s + 0.741·11-s − 3.78·12-s − 3.78·13-s − 2.35·14-s − 0.258·16-s − 3.16·17-s − 6.67·18-s − 19-s + 8.35·21-s − 0.672·22-s + 0.570·23-s + 9.27·24-s + 3.43·26-s + 14.0·27-s − 3.05·28-s + 6·29-s + 5.83·31-s − 5.52·32-s + 2.38·33-s + 2.87·34-s + ⋯ |
L(s) = 1 | − 0.641·2-s + 1.85·3-s − 0.588·4-s − 1.19·6-s + 0.981·7-s + 1.01·8-s + 2.45·9-s + 0.223·11-s − 1.09·12-s − 1.05·13-s − 0.629·14-s − 0.0647·16-s − 0.768·17-s − 1.57·18-s − 0.229·19-s + 1.82·21-s − 0.143·22-s + 0.119·23-s + 1.89·24-s + 0.673·26-s + 2.69·27-s − 0.577·28-s + 1.11·29-s + 1.04·31-s − 0.977·32-s + 0.415·33-s + 0.492·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.788661775\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.788661775\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 2 | \( 1 + 0.906T + 2T^{2} \) |
| 3 | \( 1 - 3.21T + 3T^{2} \) |
| 7 | \( 1 - 2.59T + 7T^{2} \) |
| 11 | \( 1 - 0.741T + 11T^{2} \) |
| 13 | \( 1 + 3.78T + 13T^{2} \) |
| 17 | \( 1 + 3.16T + 17T^{2} \) |
| 23 | \( 1 - 0.570T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 5.83T + 31T^{2} \) |
| 37 | \( 1 + 1.40T + 37T^{2} \) |
| 41 | \( 1 + 3.83T + 41T^{2} \) |
| 43 | \( 1 + 2.59T + 43T^{2} \) |
| 47 | \( 1 - 5.08T + 47T^{2} \) |
| 53 | \( 1 + 0.160T + 53T^{2} \) |
| 59 | \( 1 - 8.35T + 59T^{2} \) |
| 61 | \( 1 + 8.57T + 61T^{2} \) |
| 67 | \( 1 + 14.8T + 67T^{2} \) |
| 71 | \( 1 - 3.64T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 - 1.83T + 79T^{2} \) |
| 83 | \( 1 + 4.19T + 83T^{2} \) |
| 89 | \( 1 + 16.9T + 89T^{2} \) |
| 97 | \( 1 + 3.78T + 97T^{2} \) |
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
−10.51526324969677858696240230855, −9.863383752463894461799087123815, −8.981015553007399735651398476042, −8.451633327098043102856435875597, −7.79435350009868441261061293749, −6.95501002832430837916556453781, −4.81723383056450431016870446997, −4.21181576347591212337222857738, −2.73889190584271940849123491468, −1.57096617156208095214207025588,
1.57096617156208095214207025588, 2.73889190584271940849123491468, 4.21181576347591212337222857738, 4.81723383056450431016870446997, 6.95501002832430837916556453781, 7.79435350009868441261061293749, 8.451633327098043102856435875597, 8.981015553007399735651398476042, 9.863383752463894461799087123815, 10.51526324969677858696240230855