| L(s) = 1 | + 2.56·2-s + 4.56·4-s − 5-s − 7-s + 6.56·8-s − 2.56·10-s + 1.56·11-s + 0.438·13-s − 2.56·14-s + 7.68·16-s + 0.438·17-s − 7.12·19-s − 4.56·20-s + 4·22-s − 3.12·23-s + 25-s + 1.12·26-s − 4.56·28-s − 6.68·29-s + 6.56·32-s + 1.12·34-s + 35-s + 6·37-s − 18.2·38-s − 6.56·40-s − 5.12·41-s + 0.876·43-s + ⋯ |
| L(s) = 1 | + 1.81·2-s + 2.28·4-s − 0.447·5-s − 0.377·7-s + 2.31·8-s − 0.810·10-s + 0.470·11-s + 0.121·13-s − 0.684·14-s + 1.92·16-s + 0.106·17-s − 1.63·19-s − 1.01·20-s + 0.852·22-s − 0.651·23-s + 0.200·25-s + 0.220·26-s − 0.862·28-s − 1.24·29-s + 1.15·32-s + 0.192·34-s + 0.169·35-s + 0.986·37-s − 2.95·38-s − 1.03·40-s − 0.800·41-s + 0.133·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.214798294\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.214798294\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 - 2.56T + 2T^{2} \) |
| 11 | \( 1 - 1.56T + 11T^{2} \) |
| 13 | \( 1 - 0.438T + 13T^{2} \) |
| 17 | \( 1 - 0.438T + 17T^{2} \) |
| 19 | \( 1 + 7.12T + 19T^{2} \) |
| 23 | \( 1 + 3.12T + 23T^{2} \) |
| 29 | \( 1 + 6.68T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 5.12T + 41T^{2} \) |
| 43 | \( 1 - 0.876T + 43T^{2} \) |
| 47 | \( 1 - 8.68T + 47T^{2} \) |
| 53 | \( 1 - 5.12T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 15.3T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 + 2.43T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 1.12T + 89T^{2} \) |
| 97 | \( 1 - 5.80T + 97T^{2} \) |
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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
−11.86440151944169430475892478897, −11.19380226043183836751872111467, −10.17828311706138448888339154271, −8.691312556879247691907924892909, −7.38252446259464115672880705546, −6.46795697375870021901540361138, −5.63415169248422340025975915050, −4.32214639586310152414841971968, −3.67902216200559470223364508823, −2.27091041523820840992761312963,
2.27091041523820840992761312963, 3.67902216200559470223364508823, 4.32214639586310152414841971968, 5.63415169248422340025975915050, 6.46795697375870021901540361138, 7.38252446259464115672880705546, 8.691312556879247691907924892909, 10.17828311706138448888339154271, 11.19380226043183836751872111467, 11.86440151944169430475892478897
