| L(s) = 1 | − 5-s − 11-s − 6·13-s + 5·17-s − 2·19-s − 23-s − 4·25-s − 4·29-s + 2·31-s − 41-s + 8·43-s − 47-s − 10·53-s + 55-s + 6·59-s − 7·61-s + 6·65-s + 3·67-s − 2·73-s − 79-s + 9·83-s − 5·85-s − 16·89-s + 2·95-s − 11·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s − 1.66·13-s + 1.21·17-s − 0.458·19-s − 0.208·23-s − 4/5·25-s − 0.742·29-s + 0.359·31-s − 0.156·41-s + 1.21·43-s − 0.145·47-s − 1.37·53-s + 0.134·55-s + 0.781·59-s − 0.896·61-s + 0.744·65-s + 0.366·67-s − 0.234·73-s − 0.112·79-s + 0.987·83-s − 0.542·85-s − 1.69·89-s + 0.205·95-s − 1.11·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4227774888\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4227774888\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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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
−12.62281028629432, −12.22677292963305, −11.82275660521816, −11.34225437166617, −10.84661382474169, −10.32242024956728, −9.850447571948741, −9.564936623158281, −9.099923564129350, −8.312677754950784, −8.000352238203943, −7.517814767217893, −7.280076876058872, −6.626804970919968, −6.007952269383064, −5.545698571506595, −5.083481765364010, −4.543805817078523, −4.052596894017883, −3.504838245402027, −2.878535243241559, −2.415657538955637, −1.809846023567047, −1.079514566012563, −0.1809720632726247,
0.1809720632726247, 1.079514566012563, 1.809846023567047, 2.415657538955637, 2.878535243241559, 3.504838245402027, 4.052596894017883, 4.543805817078523, 5.083481765364010, 5.545698571506595, 6.007952269383064, 6.626804970919968, 7.280076876058872, 7.517814767217893, 8.000352238203943, 8.312677754950784, 9.099923564129350, 9.564936623158281, 9.850447571948741, 10.32242024956728, 10.84661382474169, 11.34225437166617, 11.82275660521816, 12.22677292963305, 12.62281028629432
