| L(s) = 1 | + 2.41·5-s − 4.72·7-s − 6.01·13-s − 0.890·17-s − 3.31·19-s − 0.204·23-s + 0.822·25-s + 9.57·29-s − 9.12·31-s − 11.4·35-s + 1.44·37-s − 3.54·41-s − 7.79·43-s + 9.17·47-s + 15.3·49-s + 4.64·53-s + 0.989·59-s + 6.43·61-s − 14.5·65-s + 6.17·67-s − 6.80·71-s + 3.86·73-s − 4.53·79-s + 9.99·83-s − 2.14·85-s + 0.930·89-s + 28.4·91-s + ⋯ |
| L(s) = 1 | + 1.07·5-s − 1.78·7-s − 1.66·13-s − 0.216·17-s − 0.760·19-s − 0.0427·23-s + 0.164·25-s + 1.77·29-s − 1.63·31-s − 1.92·35-s + 0.236·37-s − 0.553·41-s − 1.18·43-s + 1.33·47-s + 2.19·49-s + 0.638·53-s + 0.128·59-s + 0.823·61-s − 1.79·65-s + 0.754·67-s − 0.807·71-s + 0.452·73-s − 0.510·79-s + 1.09·83-s − 0.233·85-s + 0.0986·89-s + 2.98·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.188610581\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.188610581\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 2.41T + 5T^{2} \) |
| 7 | \( 1 + 4.72T + 7T^{2} \) |
| 13 | \( 1 + 6.01T + 13T^{2} \) |
| 17 | \( 1 + 0.890T + 17T^{2} \) |
| 19 | \( 1 + 3.31T + 19T^{2} \) |
| 23 | \( 1 + 0.204T + 23T^{2} \) |
| 29 | \( 1 - 9.57T + 29T^{2} \) |
| 31 | \( 1 + 9.12T + 31T^{2} \) |
| 37 | \( 1 - 1.44T + 37T^{2} \) |
| 41 | \( 1 + 3.54T + 41T^{2} \) |
| 43 | \( 1 + 7.79T + 43T^{2} \) |
| 47 | \( 1 - 9.17T + 47T^{2} \) |
| 53 | \( 1 - 4.64T + 53T^{2} \) |
| 59 | \( 1 - 0.989T + 59T^{2} \) |
| 61 | \( 1 - 6.43T + 61T^{2} \) |
| 67 | \( 1 - 6.17T + 67T^{2} \) |
| 71 | \( 1 + 6.80T + 71T^{2} \) |
| 73 | \( 1 - 3.86T + 73T^{2} \) |
| 79 | \( 1 + 4.53T + 79T^{2} \) |
| 83 | \( 1 - 9.99T + 83T^{2} \) |
| 89 | \( 1 - 0.930T + 89T^{2} \) |
| 97 | \( 1 - 2.92T + 97T^{2} \) |
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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
−7.57955749793332210667818022906, −6.83923615747183376337539650701, −6.52043679556453583589326630039, −5.74925837762147194084984353748, −5.14629418716363981386196303542, −4.23623176677264670577991202176, −3.32452589758957434850201753313, −2.55643012840940318600159762868, −2.02319227613603208395108320155, −0.49399035322451527059821553125,
0.49399035322451527059821553125, 2.02319227613603208395108320155, 2.55643012840940318600159762868, 3.32452589758957434850201753313, 4.23623176677264670577991202176, 5.14629418716363981386196303542, 5.74925837762147194084984353748, 6.52043679556453583589326630039, 6.83923615747183376337539650701, 7.57955749793332210667818022906
