L(s) = 1 | + 5-s + 11-s − 13-s − 7·17-s + 2·19-s − 3·23-s + 25-s − 6·29-s + 8·31-s − 37-s − 9·41-s + 2·43-s − 8·47-s + 6·53-s + 55-s − 3·59-s + 14·61-s − 65-s + 15·67-s + 8·71-s − 73-s − 79-s − 8·83-s − 7·85-s − 15·89-s + 2·95-s − 3·97-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.301·11-s − 0.277·13-s − 1.69·17-s + 0.458·19-s − 0.625·23-s + 1/5·25-s − 1.11·29-s + 1.43·31-s − 0.164·37-s − 1.40·41-s + 0.304·43-s − 1.16·47-s + 0.824·53-s + 0.134·55-s − 0.390·59-s + 1.79·61-s − 0.124·65-s + 1.83·67-s + 0.949·71-s − 0.117·73-s − 0.112·79-s − 0.878·83-s − 0.759·85-s − 1.58·89-s + 0.205·95-s − 0.304·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.553628555\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.553628555\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 15 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−13.08277581841704, −12.44348927139096, −12.02015193503476, −11.42058727210735, −11.21174188517557, −10.64013682223930, −9.913087493263745, −9.800548748969185, −9.276645773658152, −8.618130848952298, −8.332380707011875, −7.834422981896274, −6.962148235698441, −6.788090639230358, −6.398019905092432, −5.577171095747304, −5.317605890962840, −4.657813018973675, −4.121721218133559, −3.683207890129475, −2.901596264331054, −2.356285426521097, −1.887267606608970, −1.221580132761691, −0.3424926831052709,
0.3424926831052709, 1.221580132761691, 1.887267606608970, 2.356285426521097, 2.901596264331054, 3.683207890129475, 4.121721218133559, 4.657813018973675, 5.317605890962840, 5.577171095747304, 6.398019905092432, 6.788090639230358, 6.962148235698441, 7.834422981896274, 8.332380707011875, 8.618130848952298, 9.276645773658152, 9.800548748969185, 9.913087493263745, 10.64013682223930, 11.21174188517557, 11.42058727210735, 12.02015193503476, 12.44348927139096, 13.08277581841704