| L(s) = 1 | − 3·3-s − 9.84·5-s + 23.6·7-s + 9·9-s + 8.72·11-s + 29.5·15-s + 58.1·17-s − 112.·19-s − 70.8·21-s + 118.·23-s − 28.0·25-s − 27·27-s − 27.1·29-s + 119.·31-s − 26.1·33-s − 232.·35-s + 286.·37-s + 218.·41-s + 281.·43-s − 88.6·45-s + 451.·47-s + 215.·49-s − 174.·51-s − 581.·53-s − 85.9·55-s + 337.·57-s − 775.·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.880·5-s + 1.27·7-s + 0.333·9-s + 0.239·11-s + 0.508·15-s + 0.829·17-s − 1.35·19-s − 0.736·21-s + 1.07·23-s − 0.224·25-s − 0.192·27-s − 0.173·29-s + 0.692·31-s − 0.138·33-s − 1.12·35-s + 1.27·37-s + 0.833·41-s + 0.999·43-s − 0.293·45-s + 1.40·47-s + 0.628·49-s − 0.478·51-s − 1.50·53-s − 0.210·55-s + 0.783·57-s − 1.71·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.707383731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.707383731\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 9.84T + 125T^{2} \) |
| 7 | \( 1 - 23.6T + 343T^{2} \) |
| 11 | \( 1 - 8.72T + 1.33e3T^{2} \) |
| 17 | \( 1 - 58.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 112.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 118.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 27.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 119.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 286.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 218.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 281.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 451.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 581.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 775.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 35.1T + 2.26e5T^{2} \) |
| 67 | \( 1 + 845.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 30.2T + 3.57e5T^{2} \) |
| 73 | \( 1 + 276.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 120.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 954.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 669.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 343.T + 9.12e5T^{2} \) |
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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
−8.672048698875634077419633050148, −7.81525583830945662072308227255, −7.49290842410952858598689985773, −6.35478493289918042891549826275, −5.59338193541636028555139773470, −4.49659593857911956273291274901, −4.25982772685953699569851318748, −2.90483768991600953891758898008, −1.61585216375620782612002334652, −0.65143882006372289151996043790,
0.65143882006372289151996043790, 1.61585216375620782612002334652, 2.90483768991600953891758898008, 4.25982772685953699569851318748, 4.49659593857911956273291274901, 5.59338193541636028555139773470, 6.35478493289918042891549826275, 7.49290842410952858598689985773, 7.81525583830945662072308227255, 8.672048698875634077419633050148
