| L(s) = 1 | − 5.72·3-s + 12.6·7-s + 5.77·9-s − 21.9·11-s − 37.2·13-s − 62.5·17-s − 137.·19-s − 72.2·21-s + 23·23-s + 121.·27-s − 71.8·29-s − 219.·31-s + 125.·33-s + 296.·37-s + 213.·39-s − 305.·41-s − 140.·43-s + 185.·47-s − 183.·49-s + 358.·51-s − 312.·53-s + 787.·57-s − 161.·59-s + 835.·61-s + 72.8·63-s + 24.5·67-s − 131.·69-s + ⋯ |
| L(s) = 1 | − 1.10·3-s + 0.681·7-s + 0.213·9-s − 0.602·11-s − 0.795·13-s − 0.892·17-s − 1.66·19-s − 0.750·21-s + 0.208·23-s + 0.866·27-s − 0.460·29-s − 1.27·31-s + 0.663·33-s + 1.31·37-s + 0.876·39-s − 1.16·41-s − 0.497·43-s + 0.575·47-s − 0.536·49-s + 0.983·51-s − 0.809·53-s + 1.83·57-s − 0.356·59-s + 1.75·61-s + 0.145·63-s + 0.0446·67-s − 0.229·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4409110678\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4409110678\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 + 5.72T + 27T^{2} \) |
| 7 | \( 1 - 12.6T + 343T^{2} \) |
| 11 | \( 1 + 21.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 37.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 62.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 137.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 71.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 219.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 296.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 305.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 140.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 185.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 312.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 161.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 835.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 24.5T + 3.00e5T^{2} \) |
| 71 | \( 1 + 973.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 343.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.21e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 749.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 811.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.02e3T + 9.12e5T^{2} \) |
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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
−8.574355608715721736946910834419, −7.894140619788950591688151681788, −6.94254094538361728095960502751, −6.31087977903949019421354922473, −5.39051046180898439826502517568, −4.86410614229300823997892146425, −4.07215054351940112943059584966, −2.63469635550856883830500519978, −1.76269930931831973926137615279, −0.30284011569512761445940855976,
0.30284011569512761445940855976, 1.76269930931831973926137615279, 2.63469635550856883830500519978, 4.07215054351940112943059584966, 4.86410614229300823997892146425, 5.39051046180898439826502517568, 6.31087977903949019421354922473, 6.94254094538361728095960502751, 7.894140619788950591688151681788, 8.574355608715721736946910834419
