| L(s) = 1 | + 3-s − 5-s + 3.82·7-s + 9-s + 5.41·11-s + 0.585·13-s − 15-s + 8.07·17-s − 2.24·19-s + 3.82·21-s + 23-s + 25-s + 27-s − 6.41·29-s + 9.82·31-s + 5.41·33-s − 3.82·35-s + 3·37-s + 0.585·39-s − 7.58·41-s − 6·43-s − 45-s + 11.4·47-s + 7.65·49-s + 8.07·51-s − 1.24·53-s − 5.41·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.44·7-s + 0.333·9-s + 1.63·11-s + 0.162·13-s − 0.258·15-s + 1.95·17-s − 0.514·19-s + 0.835·21-s + 0.208·23-s + 0.200·25-s + 0.192·27-s − 1.19·29-s + 1.76·31-s + 0.942·33-s − 0.647·35-s + 0.493·37-s + 0.0938·39-s − 1.18·41-s − 0.914·43-s − 0.149·45-s + 1.66·47-s + 1.09·49-s + 1.13·51-s − 0.170·53-s − 0.730·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.471199166\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.471199166\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 3.82T + 7T^{2} \) |
| 11 | \( 1 - 5.41T + 11T^{2} \) |
| 13 | \( 1 - 0.585T + 13T^{2} \) |
| 17 | \( 1 - 8.07T + 17T^{2} \) |
| 19 | \( 1 + 2.24T + 19T^{2} \) |
| 29 | \( 1 + 6.41T + 29T^{2} \) |
| 31 | \( 1 - 9.82T + 31T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 + 7.58T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 + 1.24T + 53T^{2} \) |
| 59 | \( 1 + 12.8T + 59T^{2} \) |
| 61 | \( 1 - 2.58T + 61T^{2} \) |
| 67 | \( 1 + 5.48T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 + 1.75T + 73T^{2} \) |
| 79 | \( 1 + 7.31T + 79T^{2} \) |
| 83 | \( 1 - 0.0710T + 83T^{2} \) |
| 89 | \( 1 + 18.1T + 89T^{2} \) |
| 97 | \( 1 + 6.82T + 97T^{2} \) |
| 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
−8.186838891654689193585792915699, −7.61038407133851102709161456071, −6.89762682289004846920100763336, −6.01062516328945728729552727363, −5.15100354273576273189962222241, −4.32497321234493354863663967334, −3.79033189161617006723110801665, −2.90925675270995681314017191979, −1.64256685714035588528212738351, −1.13493705019001449289508767985,
1.13493705019001449289508767985, 1.64256685714035588528212738351, 2.90925675270995681314017191979, 3.79033189161617006723110801665, 4.32497321234493354863663967334, 5.15100354273576273189962222241, 6.01062516328945728729552727363, 6.89762682289004846920100763336, 7.61038407133851102709161456071, 8.186838891654689193585792915699
