L(s) = 1 | + (0.587 + 0.809i)2-s + (−0.309 + 0.951i)4-s + (−0.951 + 0.309i)7-s + (−0.951 + 0.309i)8-s + (−0.587 + 0.809i)9-s + (−0.809 + 0.587i)11-s + (−0.809 − 0.587i)14-s + (−0.809 − 0.587i)16-s − 18-s + (−0.951 − 0.309i)22-s + 0.618·23-s + (−0.951 − 0.309i)25-s − i·28-s + (1.76 + 0.896i)29-s − i·32-s + ⋯ |
L(s) = 1 | + (0.587 + 0.809i)2-s + (−0.309 + 0.951i)4-s + (−0.951 + 0.309i)7-s + (−0.951 + 0.309i)8-s + (−0.587 + 0.809i)9-s + (−0.809 + 0.587i)11-s + (−0.809 − 0.587i)14-s + (−0.809 − 0.587i)16-s − 18-s + (−0.951 − 0.309i)22-s + 0.618·23-s + (−0.951 − 0.309i)25-s − i·28-s + (1.76 + 0.896i)29-s − i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 - 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 - 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9272104014\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9272104014\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.587 - 0.809i)T \) |
| 7 | \( 1 + (0.951 - 0.309i)T \) |
| 11 | \( 1 + (0.809 - 0.587i)T \) |
good | 3 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 5 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 13 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 17 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 23 | \( 1 - 0.618T + T^{2} \) |
| 29 | \( 1 + (-1.76 - 0.896i)T + (0.587 + 0.809i)T^{2} \) |
| 31 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (-1.58 - 0.809i)T + (0.587 + 0.809i)T^{2} \) |
| 41 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 + (1.39 - 1.39i)T - iT^{2} \) |
| 47 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (0.309 - 0.0489i)T + (0.951 - 0.309i)T^{2} \) |
| 59 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 61 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 67 | \( 1 + (-0.221 + 0.221i)T - iT^{2} \) |
| 71 | \( 1 + (-0.951 + 0.690i)T + (0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.951 - 0.690i)T + (0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
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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
−10.12008123850530905777183922173, −9.395666181662086426095995285718, −8.331140122319446606155146061179, −7.87992424024122423551650651697, −6.81316501429242607662673818150, −6.16856019227940263209293669863, −5.20609638373333316226101143781, −4.57686055612469915937870619743, −3.19759400826585155381332593633, −2.51984883279180779872981278554,
0.63827619638058706844403075378, 2.51594484645294246523006693661, 3.26992477077742903244097839656, 4.09359917909233905744939867340, 5.30168034799437254948189080582, 6.07755807799739550470581666316, 6.73759809063434158071401790212, 8.060764475009186772617559993690, 9.003948958613060925580530869523, 9.732224409895372629174613210341