L(s) = 1 | + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s + (−1.30 + 0.951i)13-s + (−0.809 + 0.587i)16-s + (−0.190 + 0.587i)17-s − 18-s − 1.61·26-s + (0.190 + 0.587i)29-s − 32-s + (−0.5 + 0.363i)34-s + (−0.809 − 0.587i)36-s + (0.5 − 0.363i)37-s + (1.30 − 0.951i)41-s + ⋯ |
L(s) = 1 | + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s + (−1.30 + 0.951i)13-s + (−0.809 + 0.587i)16-s + (−0.190 + 0.587i)17-s − 18-s − 1.61·26-s + (0.190 + 0.587i)29-s − 32-s + (−0.5 + 0.363i)34-s + (−0.809 − 0.587i)36-s + (0.5 − 0.363i)37-s + (1.30 − 0.951i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.469552068\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.469552068\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.809 - 0.587i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 17 | \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 29 | \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 71 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 97 | \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
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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
−9.042816345950848547295866328787, −8.630621937921865682569906492123, −7.52734069540088603883847355445, −7.22517112789986821950921384255, −6.14925205868528148928978132087, −5.54085708956403222787278723645, −4.67505230967454342017837014212, −4.02780278679253793547775505645, −2.80374193750276773034243885836, −2.13063426063701713359973488934,
0.69999218743651607262078314993, 2.39227651126043525990292547829, 2.91495901844413406656902309694, 3.92753396762493846777370611865, 4.88963896468221950264321865224, 5.52362361463056173792463037062, 6.29719705657601986265894002770, 7.13921509636421032879954577086, 8.009065912854725802811762953853, 9.012778273656195249312756787179