| L(s) = 1 | − 2·3-s − 5-s + 9-s − 11-s − 3·13-s + 2·15-s − 2·17-s − 5·19-s + 7·23-s + 25-s + 4·27-s − 6·29-s + 4·31-s + 2·33-s − 5·37-s + 6·39-s − 5·41-s + 6·43-s − 45-s − 9·47-s + 4·51-s + 11·53-s + 55-s + 10·57-s + 8·59-s − 12·61-s + 3·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s + 1/3·9-s − 0.301·11-s − 0.832·13-s + 0.516·15-s − 0.485·17-s − 1.14·19-s + 1.45·23-s + 1/5·25-s + 0.769·27-s − 1.11·29-s + 0.718·31-s + 0.348·33-s − 0.821·37-s + 0.960·39-s − 0.780·41-s + 0.914·43-s − 0.149·45-s − 1.31·47-s + 0.560·51-s + 1.51·53-s + 0.134·55-s + 1.32·57-s + 1.04·59-s − 1.53·61-s + 0.372·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6227376888\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6227376888\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−9.120494019087879646941493450871, −8.447359674641968247307299532517, −7.41175041159849215065524714179, −6.78703353203608464553115045216, −5.99797434225824589956437343584, −5.05403415737104360436911227846, −4.59475108981645282369545843095, −3.36085533604321633795096739870, −2.17010984041333828186019700181, −0.53684007895046932778022767435,
0.53684007895046932778022767435, 2.17010984041333828186019700181, 3.36085533604321633795096739870, 4.59475108981645282369545843095, 5.05403415737104360436911227846, 5.99797434225824589956437343584, 6.78703353203608464553115045216, 7.41175041159849215065524714179, 8.447359674641968247307299532517, 9.120494019087879646941493450871
