| L(s) = 1 | + 3·3-s − 4.58·5-s + 9·9-s + 6.48·11-s − 45.2·13-s − 13.7·15-s − 81.5·17-s − 5.05·19-s − 106.·23-s − 103.·25-s + 27·27-s − 268.·29-s + 292.·31-s + 19.4·33-s + 114.·37-s − 135.·39-s + 161.·41-s + 471.·43-s − 41.2·45-s + 346.·47-s − 244.·51-s + 405.·53-s − 29.7·55-s − 15.1·57-s − 253.·59-s + 751.·61-s + 207.·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.410·5-s + 0.333·9-s + 0.177·11-s − 0.964·13-s − 0.236·15-s − 1.16·17-s − 0.0610·19-s − 0.963·23-s − 0.831·25-s + 0.192·27-s − 1.71·29-s + 1.69·31-s + 0.102·33-s + 0.509·37-s − 0.556·39-s + 0.615·41-s + 1.67·43-s − 0.136·45-s + 1.07·47-s − 0.671·51-s + 1.05·53-s − 0.0729·55-s − 0.0352·57-s − 0.559·59-s + 1.57·61-s + 0.395·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.867755349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.867755349\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 4.58T + 125T^{2} \) |
| 11 | \( 1 - 6.48T + 1.33e3T^{2} \) |
| 13 | \( 1 + 45.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 81.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 5.05T + 6.85e3T^{2} \) |
| 23 | \( 1 + 106.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 268.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 292.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 114.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 161.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 471.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 346.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 405.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 253.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 751.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 11.6T + 3.00e5T^{2} \) |
| 71 | \( 1 - 681.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 685.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 0.264T + 4.93e5T^{2} \) |
| 83 | \( 1 + 437.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 58.5T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.28e3T + 9.12e5T^{2} \) |
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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
−8.633317885479656665747045176627, −7.78359990256618150708105149294, −7.33111070285033705664592543350, −6.37655145753054189368115001657, −5.50245667642303507380810097949, −4.30305942888510030851443219707, −3.98662607110675946174460996874, −2.65217991889846832478146604881, −2.04550379913908738161928337724, −0.56710428894951888202403424681,
0.56710428894951888202403424681, 2.04550379913908738161928337724, 2.65217991889846832478146604881, 3.98662607110675946174460996874, 4.30305942888510030851443219707, 5.50245667642303507380810097949, 6.37655145753054189368115001657, 7.33111070285033705664592543350, 7.78359990256618150708105149294, 8.633317885479656665747045176627
