| L(s) = 1 | + 2-s − 3.06·3-s + 4-s − 1.76·5-s − 3.06·6-s + 8-s + 6.41·9-s − 1.76·10-s − 5.93·11-s − 3.06·12-s − 1.76·13-s + 5.41·15-s + 16-s − 1.30·17-s + 6.41·18-s − 3.86·19-s − 1.76·20-s − 5.93·22-s + 4.52·23-s − 3.06·24-s − 1.88·25-s − 1.76·26-s − 10.4·27-s + 29-s + 5.41·30-s − 0.504·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.77·3-s + 0.5·4-s − 0.788·5-s − 1.25·6-s + 0.353·8-s + 2.13·9-s − 0.557·10-s − 1.78·11-s − 0.885·12-s − 0.489·13-s + 1.39·15-s + 0.250·16-s − 0.316·17-s + 1.51·18-s − 0.887·19-s − 0.394·20-s − 1.26·22-s + 0.942·23-s − 0.626·24-s − 0.377·25-s − 0.345·26-s − 2.01·27-s + 0.185·29-s + 0.987·30-s − 0.0905·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6582148814\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6582148814\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 3.06T + 3T^{2} \) |
| 5 | \( 1 + 1.76T + 5T^{2} \) |
| 11 | \( 1 + 5.93T + 11T^{2} \) |
| 13 | \( 1 + 1.76T + 13T^{2} \) |
| 17 | \( 1 + 1.30T + 17T^{2} \) |
| 19 | \( 1 + 3.86T + 19T^{2} \) |
| 23 | \( 1 - 4.52T + 23T^{2} \) |
| 31 | \( 1 + 0.504T + 31T^{2} \) |
| 37 | \( 1 + 5.34T + 37T^{2} \) |
| 41 | \( 1 - 9.03T + 41T^{2} \) |
| 43 | \( 1 + 5.41T + 43T^{2} \) |
| 47 | \( 1 - 4.03T + 47T^{2} \) |
| 53 | \( 1 + 3.41T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 1.25T + 61T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 + 6.29T + 71T^{2} \) |
| 73 | \( 1 - 1.30T + 73T^{2} \) |
| 79 | \( 1 + 1.41T + 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 + 0.295T + 89T^{2} \) |
| 97 | \( 1 - 18.0T + 97T^{2} \) |
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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
−8.635070758791701061664633272084, −7.56352921116222478858018978455, −7.24977719882818542595791903343, −6.25228786994763923193207994559, −5.65383847825013940275738167518, −4.78482431548293285136793224764, −4.55955623812166782964169344904, −3.31168337844983040532412031550, −2.11009001612084157111746428858, −0.46792106080663830312129838989,
0.46792106080663830312129838989, 2.11009001612084157111746428858, 3.31168337844983040532412031550, 4.55955623812166782964169344904, 4.78482431548293285136793224764, 5.65383847825013940275738167518, 6.25228786994763923193207994559, 7.24977719882818542595791903343, 7.56352921116222478858018978455, 8.635070758791701061664633272084
