L(s) = 1 | + (−0.453 − 0.891i)2-s + (−0.587 + 0.809i)4-s + (0.156 − 0.987i)5-s + (0.987 + 0.156i)8-s + (−0.951 + 0.309i)10-s + (0.278 + 0.142i)13-s + (−0.309 − 0.951i)16-s + (0.297 − 1.87i)17-s + (0.707 + 0.707i)20-s + (−0.951 − 0.309i)25-s − 0.312i·26-s + (0.734 + 0.533i)29-s + (−0.707 + 0.707i)32-s + (−1.80 + 0.587i)34-s + (0.809 − 1.58i)37-s + ⋯ |
L(s) = 1 | + (−0.453 − 0.891i)2-s + (−0.587 + 0.809i)4-s + (0.156 − 0.987i)5-s + (0.987 + 0.156i)8-s + (−0.951 + 0.309i)10-s + (0.278 + 0.142i)13-s + (−0.309 − 0.951i)16-s + (0.297 − 1.87i)17-s + (0.707 + 0.707i)20-s + (−0.951 − 0.309i)25-s − 0.312i·26-s + (0.734 + 0.533i)29-s + (−0.707 + 0.707i)32-s + (−1.80 + 0.587i)34-s + (0.809 − 1.58i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.356 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.356 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7627811487\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7627811487\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.453 + 0.891i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.156 + 0.987i)T \) |
good | 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.278 - 0.142i)T + (0.587 + 0.809i)T^{2} \) |
| 17 | \( 1 + (-0.297 + 1.87i)T + (-0.951 - 0.309i)T^{2} \) |
| 19 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 29 | \( 1 + (-0.734 - 0.533i)T + (0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.809 + 1.58i)T + (-0.587 - 0.809i)T^{2} \) |
| 41 | \( 1 + (1.87 - 0.610i)T + (0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 53 | \( 1 + (0.183 + 1.16i)T + (-0.951 + 0.309i)T^{2} \) |
| 59 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 71 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.896 - 1.76i)T + (-0.587 + 0.809i)T^{2} \) |
| 79 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 89 | \( 1 + (0.550 - 1.69i)T + (-0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (-0.142 - 0.896i)T + (-0.951 + 0.309i)T^{2} \) |
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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
−9.820990012491905754908140864064, −9.407601358549081031595352006472, −8.578731012643883009965561606997, −7.83682480365898620927376456531, −6.81654864146136128318978113403, −5.32812962155804954780888621141, −4.67634678205939465195635620613, −3.53458629210038054055432311910, −2.33432413161876626903061854070, −0.968469686603993211034802268358,
1.73339998890753683225415096372, 3.34006237433020444579974751399, 4.45996800367113574001194358652, 5.76280529887054441818396383952, 6.31843822283589925636486984091, 7.11580315275210090739241463723, 8.095988553510903370117928663792, 8.622485955073829276506194713862, 9.915679063825848240511083519135, 10.27404933109684301280784186878