| L(s) = 1 | − 0.845·3-s + 5-s + 2.14·7-s − 2.28·9-s − 1.83·11-s − 2.76·13-s − 0.845·15-s − 6.31·17-s + 1.00·19-s − 1.81·21-s + 5.38·23-s + 25-s + 4.46·27-s + 5.38·29-s + 3.18·31-s + 1.55·33-s + 2.14·35-s − 37-s + 2.33·39-s − 1.83·41-s + 1.11·43-s − 2.28·45-s + 7.02·47-s − 2.41·49-s + 5.33·51-s + 13.9·53-s − 1.83·55-s + ⋯ |
| L(s) = 1 | − 0.487·3-s + 0.447·5-s + 0.809·7-s − 0.761·9-s − 0.553·11-s − 0.767·13-s − 0.218·15-s − 1.53·17-s + 0.231·19-s − 0.395·21-s + 1.12·23-s + 0.200·25-s + 0.859·27-s + 1.00·29-s + 0.572·31-s + 0.270·33-s + 0.362·35-s − 0.164·37-s + 0.374·39-s − 0.286·41-s + 0.170·43-s − 0.340·45-s + 1.02·47-s − 0.344·49-s + 0.747·51-s + 1.91·53-s − 0.247·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.462301903\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.462301903\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 + 0.845T + 3T^{2} \) |
| 7 | \( 1 - 2.14T + 7T^{2} \) |
| 11 | \( 1 + 1.83T + 11T^{2} \) |
| 13 | \( 1 + 2.76T + 13T^{2} \) |
| 17 | \( 1 + 6.31T + 17T^{2} \) |
| 19 | \( 1 - 1.00T + 19T^{2} \) |
| 23 | \( 1 - 5.38T + 23T^{2} \) |
| 29 | \( 1 - 5.38T + 29T^{2} \) |
| 31 | \( 1 - 3.18T + 31T^{2} \) |
| 41 | \( 1 + 1.83T + 41T^{2} \) |
| 43 | \( 1 - 1.11T + 43T^{2} \) |
| 47 | \( 1 - 7.02T + 47T^{2} \) |
| 53 | \( 1 - 13.9T + 53T^{2} \) |
| 59 | \( 1 - 2.21T + 59T^{2} \) |
| 61 | \( 1 - 8.69T + 61T^{2} \) |
| 67 | \( 1 - 15.0T + 67T^{2} \) |
| 71 | \( 1 - 6.27T + 71T^{2} \) |
| 73 | \( 1 + 7.15T + 73T^{2} \) |
| 79 | \( 1 - 6.70T + 79T^{2} \) |
| 83 | \( 1 + 8.35T + 83T^{2} \) |
| 89 | \( 1 - 2.59T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−8.652670389379222817477626098139, −8.156227627995209545889864796788, −7.06042199503260655937219784772, −6.55429518373547935351282841707, −5.43362075314689758605319301072, −5.09498721469517367504986536248, −4.26236822829764985753743501001, −2.81182633985447652742022627245, −2.21041187553481572909436610966, −0.74346679533909908159908447173,
0.74346679533909908159908447173, 2.21041187553481572909436610966, 2.81182633985447652742022627245, 4.26236822829764985753743501001, 5.09498721469517367504986536248, 5.43362075314689758605319301072, 6.55429518373547935351282841707, 7.06042199503260655937219784772, 8.156227627995209545889864796788, 8.652670389379222817477626098139
