| L(s) = 1 | + 2.88·2-s + 2.89·3-s + 0.299·4-s + 8.32·6-s − 22.1·8-s − 18.6·9-s − 46.4·11-s + 0.865·12-s + 31.0·13-s − 66.3·16-s + 61.8·17-s − 53.7·18-s + 24.6·19-s − 133.·22-s + 154.·23-s − 64.1·24-s + 89.4·26-s − 131.·27-s + 200.·29-s + 129.·31-s − 13.5·32-s − 134.·33-s + 178.·34-s − 5.58·36-s + 77.9·37-s + 70.9·38-s + 89.7·39-s + ⋯ |
| L(s) = 1 | + 1.01·2-s + 0.556·3-s + 0.0374·4-s + 0.566·6-s − 0.980·8-s − 0.690·9-s − 1.27·11-s + 0.0208·12-s + 0.662·13-s − 1.03·16-s + 0.882·17-s − 0.703·18-s + 0.297·19-s − 1.29·22-s + 1.40·23-s − 0.545·24-s + 0.674·26-s − 0.940·27-s + 1.28·29-s + 0.748·31-s − 0.0748·32-s − 0.708·33-s + 0.898·34-s − 0.0258·36-s + 0.346·37-s + 0.302·38-s + 0.368·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.377412725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.377412725\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.88T + 8T^{2} \) |
| 3 | \( 1 - 2.89T + 27T^{2} \) |
| 11 | \( 1 + 46.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 31.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 61.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 24.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 154.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 200.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 129.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 77.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 235.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 278.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 368.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 169.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 691.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 696.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 2.33T + 3.00e5T^{2} \) |
| 71 | \( 1 + 866.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 752.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 842.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.44e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.43e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 23.4T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.240307906890348797043357718371, −8.459953721711134951496109038640, −7.86963502568768731922682445376, −6.67886077826207204683604488272, −5.65605088819469030294744711185, −5.17839026223458085749964505737, −4.11870755244066184986582619141, −3.04853528192477598243290769731, −2.67203049278395694197154159790, −0.77502464694909192650427165752,
0.77502464694909192650427165752, 2.67203049278395694197154159790, 3.04853528192477598243290769731, 4.11870755244066184986582619141, 5.17839026223458085749964505737, 5.65605088819469030294744711185, 6.67886077826207204683604488272, 7.86963502568768731922682445376, 8.459953721711134951496109038640, 9.240307906890348797043357718371
