L(s) = 1 | + 3-s + 5-s − 7-s + 9-s − 2·13-s + 15-s + 2·17-s + 4·19-s − 21-s + 25-s + 27-s − 2·29-s − 4·31-s − 35-s − 2·37-s − 2·39-s − 2·41-s + 8·43-s + 45-s + 4·47-s + 49-s + 2·51-s + 2·53-s + 4·57-s + 12·59-s + 14·61-s − 63-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.554·13-s + 0.258·15-s + 0.485·17-s + 0.917·19-s − 0.218·21-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.718·31-s − 0.169·35-s − 0.328·37-s − 0.320·39-s − 0.312·41-s + 1.21·43-s + 0.149·45-s + 0.583·47-s + 1/7·49-s + 0.280·51-s + 0.274·53-s + 0.529·57-s + 1.56·59-s + 1.79·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.668240631\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.668240631\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.992916977932944936797672340593, −7.19252897173929783830112485704, −6.83682797043288965847256052154, −5.64685920182030477581597271333, −5.36850871739976331232751802839, −4.24324963828040919469234293294, −3.51938253594369657715591638086, −2.72213648359350533939738308476, −1.96365949268556113008382225940, −0.819774884754905808547766476547,
0.819774884754905808547766476547, 1.96365949268556113008382225940, 2.72213648359350533939738308476, 3.51938253594369657715591638086, 4.24324963828040919469234293294, 5.36850871739976331232751802839, 5.64685920182030477581597271333, 6.83682797043288965847256052154, 7.19252897173929783830112485704, 7.992916977932944936797672340593