L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s − 3.41·11-s + 16-s + 1.41·17-s + 2.82·19-s − 20-s + 3.41·22-s + 0.828·23-s + 25-s + 0.242·29-s − 9.07·31-s − 32-s − 1.41·34-s + 1.41·37-s − 2.82·38-s + 40-s + 3.17·41-s + 7.41·43-s − 3.41·44-s − 0.828·46-s − 5.07·47-s − 50-s − 13.3·53-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 1.02·11-s + 0.250·16-s + 0.342·17-s + 0.648·19-s − 0.223·20-s + 0.727·22-s + 0.172·23-s + 0.200·25-s + 0.0450·29-s − 1.62·31-s − 0.176·32-s − 0.242·34-s + 0.232·37-s − 0.458·38-s + 0.158·40-s + 0.495·41-s + 1.13·43-s − 0.514·44-s − 0.122·46-s − 0.739·47-s − 0.141·50-s − 1.82·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9413345919\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9413345919\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 3.41T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 0.828T + 23T^{2} \) |
| 29 | \( 1 - 0.242T + 29T^{2} \) |
| 31 | \( 1 + 9.07T + 31T^{2} \) |
| 37 | \( 1 - 1.41T + 37T^{2} \) |
| 41 | \( 1 - 3.17T + 41T^{2} \) |
| 43 | \( 1 - 7.41T + 43T^{2} \) |
| 47 | \( 1 + 5.07T + 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 - 14.4T + 59T^{2} \) |
| 61 | \( 1 + 0.343T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 + 5.17T + 71T^{2} \) |
| 73 | \( 1 + 3.65T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 - 3.17T + 97T^{2} \) |
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
−8.308674061208255034648243041832, −7.55436716644069849835761697745, −7.32546787190123651878832445758, −6.21449355792354880663306761654, −5.48323556856904407478938154526, −4.70358951652188454730043235120, −3.59356970997131885020495353865, −2.86485489418777190231753721933, −1.84050962798558355654630031149, −0.59589560558893302413907445030,
0.59589560558893302413907445030, 1.84050962798558355654630031149, 2.86485489418777190231753721933, 3.59356970997131885020495353865, 4.70358951652188454730043235120, 5.48323556856904407478938154526, 6.21449355792354880663306761654, 7.32546787190123651878832445758, 7.55436716644069849835761697745, 8.308674061208255034648243041832