| L(s) = 1 | + 3-s − 1.28·5-s + 7-s + 9-s − 4.20·11-s − 13-s − 1.28·15-s + 1.62·17-s + 6.33·19-s + 21-s + 2.71·23-s − 3.33·25-s + 27-s − 4.33·29-s + 7.49·31-s − 4.20·33-s − 1.28·35-s + 3.42·37-s − 39-s − 7.62·41-s + 4.91·43-s − 1.28·45-s + 1.08·47-s + 49-s + 1.62·51-s + 1.75·53-s + 5.42·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.576·5-s + 0.377·7-s + 0.333·9-s − 1.26·11-s − 0.277·13-s − 0.332·15-s + 0.394·17-s + 1.45·19-s + 0.218·21-s + 0.565·23-s − 0.667·25-s + 0.192·27-s − 0.805·29-s + 1.34·31-s − 0.732·33-s − 0.217·35-s + 0.562·37-s − 0.160·39-s − 1.19·41-s + 0.749·43-s − 0.192·45-s + 0.158·47-s + 0.142·49-s + 0.227·51-s + 0.241·53-s + 0.731·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.050370135\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.050370135\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 1.28T + 5T^{2} \) |
| 11 | \( 1 + 4.20T + 11T^{2} \) |
| 17 | \( 1 - 1.62T + 17T^{2} \) |
| 19 | \( 1 - 6.33T + 19T^{2} \) |
| 23 | \( 1 - 2.71T + 23T^{2} \) |
| 29 | \( 1 + 4.33T + 29T^{2} \) |
| 31 | \( 1 - 7.49T + 31T^{2} \) |
| 37 | \( 1 - 3.42T + 37T^{2} \) |
| 41 | \( 1 + 7.62T + 41T^{2} \) |
| 43 | \( 1 - 4.91T + 43T^{2} \) |
| 47 | \( 1 - 1.08T + 47T^{2} \) |
| 53 | \( 1 - 1.75T + 53T^{2} \) |
| 59 | \( 1 - 4.57T + 59T^{2} \) |
| 61 | \( 1 + 5.25T + 61T^{2} \) |
| 67 | \( 1 + 8.67T + 67T^{2} \) |
| 71 | \( 1 - 6.78T + 71T^{2} \) |
| 73 | \( 1 - 4.07T + 73T^{2} \) |
| 79 | \( 1 - 8.91T + 79T^{2} \) |
| 83 | \( 1 - 4.33T + 83T^{2} \) |
| 89 | \( 1 - 4.54T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−8.201893332929338816392195270301, −7.59501654568548622602585162965, −7.37298208920906662431786426508, −6.14329081718202309471318853671, −5.23029837949458159547916412883, −4.70801385152248159167044799619, −3.64171167436938381307480374898, −2.98160346739094376698668760361, −2.08709905976751273844848748931, −0.78064830568712413760106063128,
0.78064830568712413760106063128, 2.08709905976751273844848748931, 2.98160346739094376698668760361, 3.64171167436938381307480374898, 4.70801385152248159167044799619, 5.23029837949458159547916412883, 6.14329081718202309471318853671, 7.37298208920906662431786426508, 7.59501654568548622602585162965, 8.201893332929338816392195270301
