| L(s) = 1 | + (−1.32 − 0.5i)2-s + (1.50 + 1.32i)4-s + 2.64i·7-s + (−1.32 − 2.50i)8-s + 4i·11-s + (1.32 − 3.50i)14-s + (0.500 + 3.96i)16-s + (2 − 5.29i)22-s + 8i·23-s + 5·25-s + (−3.50 + 3.96i)28-s + 10.5·29-s + (1.32 − 5.50i)32-s − 6·37-s − 5.29i·43-s + (−5.29 + 6.00i)44-s + ⋯ |
| L(s) = 1 | + (−0.935 − 0.353i)2-s + (0.750 + 0.661i)4-s + 0.999i·7-s + (−0.467 − 0.883i)8-s + 1.20i·11-s + (0.353 − 0.935i)14-s + (0.125 + 0.992i)16-s + (0.426 − 1.12i)22-s + 1.66i·23-s + 25-s + (−0.661 + 0.749i)28-s + 1.96·29-s + (0.233 − 0.972i)32-s − 0.986·37-s − 0.806i·43-s + (−0.797 + 0.904i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 - 0.750i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.661 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.714463 + 0.322520i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.714463 + 0.322520i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.32 + 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
| good | 5 | \( 1 - 5T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 - 10.5T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 5.29iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 15.8iT - 67T^{2} \) |
| 71 | \( 1 + 16iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 15.8iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−12.19166573106471230483258591089, −11.16904183712607068987936832617, −10.10146721438861695000543939876, −9.337273841677232450114046862125, −8.468395675941094319345143231234, −7.40818182939887828429104272260, −6.39791320212899000378501938355, −4.92719967959972289783433168782, −3.16590581121915030195496199756, −1.83279581159740924653256896651,
0.884868169770938756035335925143, 2.96645422190522964272183663463, 4.72726421005485716529762933742, 6.21588114120258823031094461626, 6.96018890078184126814115718819, 8.191166625525540201548625739406, 8.783625229935746385680216189533, 10.16438442467027539948053982493, 10.66331584574789378433866118234, 11.58235309423722446200472603460
