| L(s) = 1 | + 2-s + 1.30·3-s + 4-s − 5-s + 1.30·6-s + 4.30·7-s + 8-s − 1.30·9-s − 10-s + 4.60·11-s + 1.30·12-s + 2.69·13-s + 4.30·14-s − 1.30·15-s + 16-s − 6.90·17-s − 1.30·18-s − 6.60·19-s − 20-s + 5.60·21-s + 4.60·22-s + 5.30·23-s + 1.30·24-s + 25-s + 2.69·26-s − 5.60·27-s + 4.30·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.752·3-s + 0.5·4-s − 0.447·5-s + 0.531·6-s + 1.62·7-s + 0.353·8-s − 0.434·9-s − 0.316·10-s + 1.38·11-s + 0.376·12-s + 0.748·13-s + 1.14·14-s − 0.336·15-s + 0.250·16-s − 1.67·17-s − 0.307·18-s − 1.51·19-s − 0.223·20-s + 1.22·21-s + 0.981·22-s + 1.10·23-s + 0.265·24-s + 0.200·25-s + 0.528·26-s − 1.07·27-s + 0.813·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.253729084\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.253729084\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 - 1.30T + 3T^{2} \) |
| 7 | \( 1 - 4.30T + 7T^{2} \) |
| 11 | \( 1 - 4.60T + 11T^{2} \) |
| 13 | \( 1 - 2.69T + 13T^{2} \) |
| 17 | \( 1 + 6.90T + 17T^{2} \) |
| 19 | \( 1 + 6.60T + 19T^{2} \) |
| 23 | \( 1 - 5.30T + 23T^{2} \) |
| 31 | \( 1 + 2.90T + 31T^{2} \) |
| 37 | \( 1 - 11.8T + 37T^{2} \) |
| 41 | \( 1 - 1.39T + 41T^{2} \) |
| 43 | \( 1 - 0.302T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 6.90T + 53T^{2} \) |
| 59 | \( 1 - 9.90T + 59T^{2} \) |
| 61 | \( 1 - 13.9T + 61T^{2} \) |
| 67 | \( 1 - 5.21T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 15.5T + 73T^{2} \) |
| 79 | \( 1 + 5.90T + 79T^{2} \) |
| 83 | \( 1 + 1.39T + 83T^{2} \) |
| 89 | \( 1 - 7.39T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−7.946087302569203386375577102021, −6.97116943588337089919848680061, −6.52590171199042252092344474882, −5.63769972244831368508353896350, −4.79105686347036940360900821447, −4.02546382747371126030608468291, −3.90095564881006231238472919068, −2.54140616672556691719845360425, −2.05776961745747128379038626732, −1.02338165307282600962929892336,
1.02338165307282600962929892336, 2.05776961745747128379038626732, 2.54140616672556691719845360425, 3.90095564881006231238472919068, 4.02546382747371126030608468291, 4.79105686347036940360900821447, 5.63769972244831368508353896350, 6.52590171199042252092344474882, 6.97116943588337089919848680061, 7.946087302569203386375577102021
