L(s) = 1 | + (−0.997 − 0.0747i)2-s + (0.988 + 0.149i)4-s + (0.825 + 0.766i)5-s + (−0.733 + 0.680i)7-s + (−0.974 − 0.222i)8-s + (−0.766 − 0.825i)10-s + (−1.11 + 1.63i)11-s + (0.781 − 0.623i)14-s + (0.955 + 0.294i)16-s + (0.702 + 0.880i)20-s + (1.23 − 1.54i)22-s + (0.0201 + 0.268i)25-s + (−0.826 + 0.563i)28-s + (−0.116 + 0.0931i)29-s + (0.510 − 0.294i)31-s + (−0.930 − 0.365i)32-s + ⋯ |
L(s) = 1 | + (−0.997 − 0.0747i)2-s + (0.988 + 0.149i)4-s + (0.825 + 0.766i)5-s + (−0.733 + 0.680i)7-s + (−0.974 − 0.222i)8-s + (−0.766 − 0.825i)10-s + (−1.11 + 1.63i)11-s + (0.781 − 0.623i)14-s + (0.955 + 0.294i)16-s + (0.702 + 0.880i)20-s + (1.23 − 1.54i)22-s + (0.0201 + 0.268i)25-s + (−0.826 + 0.563i)28-s + (−0.116 + 0.0931i)29-s + (0.510 − 0.294i)31-s + (−0.930 − 0.365i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5680191882\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5680191882\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.997 + 0.0747i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.733 - 0.680i)T \) |
good | 5 | \( 1 + (-0.825 - 0.766i)T + (0.0747 + 0.997i)T^{2} \) |
| 11 | \( 1 + (1.11 - 1.63i)T + (-0.365 - 0.930i)T^{2} \) |
| 13 | \( 1 + (0.623 - 0.781i)T^{2} \) |
| 17 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.733 + 0.680i)T^{2} \) |
| 29 | \( 1 + (0.116 - 0.0931i)T + (0.222 - 0.974i)T^{2} \) |
| 31 | \( 1 + (-0.510 + 0.294i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.955 + 0.294i)T^{2} \) |
| 41 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 43 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 47 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 53 | \( 1 + (0.149 - 0.988i)T + (-0.955 - 0.294i)T^{2} \) |
| 59 | \( 1 + (1.26 - 1.17i)T + (0.0747 - 0.997i)T^{2} \) |
| 61 | \( 1 + (0.955 - 0.294i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 73 | \( 1 + (1.94 - 0.145i)T + (0.988 - 0.149i)T^{2} \) |
| 79 | \( 1 + (0.733 - 1.26i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (1.79 + 0.865i)T + (0.623 + 0.781i)T^{2} \) |
| 89 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 97 | \( 1 - 1.86iT - T^{2} \) |
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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
−9.194786558632940740004028951218, −8.401698694233157616870046932324, −7.42231623485260068635600669613, −7.05132630287049529769910685675, −6.14129377203774278834880288003, −5.66039718549818547074063927105, −4.48565663305794500472884479674, −3.00146192521982958460583721770, −2.54419867884227787254708687140, −1.76095233273823628199164306621,
0.43277658799497628424043042551, 1.52660195330248262100242620880, 2.76793345889561694475867099910, 3.43537272943795593857295776647, 4.85506059729793360753470707882, 5.78018305105632458374499464442, 6.13688442634434483961666740305, 7.09835334488399715949391218674, 7.901261182438508034419411136782, 8.552738349758547242478512332721