| L(s) = 1 | − 3·3-s − 20.0·5-s − 32.9·7-s + 9·9-s − 43.9·11-s + 60.0·15-s − 110.·17-s + 112.·19-s + 98.8·21-s + 6.05·23-s + 275.·25-s − 27·27-s + 241.·29-s + 130.·31-s + 131.·33-s + 659.·35-s − 247.·37-s − 98.8·41-s + 360.·43-s − 180.·45-s − 3.04·47-s + 742.·49-s + 331.·51-s − 474.·53-s + 878.·55-s − 338.·57-s − 111.·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.79·5-s − 1.77·7-s + 0.333·9-s − 1.20·11-s + 1.03·15-s − 1.57·17-s + 1.36·19-s + 1.02·21-s + 0.0548·23-s + 2.20·25-s − 0.192·27-s + 1.54·29-s + 0.753·31-s + 0.694·33-s + 3.18·35-s − 1.10·37-s − 0.376·41-s + 1.27·43-s − 0.596·45-s − 0.00945·47-s + 2.16·49-s + 0.911·51-s − 1.23·53-s + 2.15·55-s − 0.786·57-s − 0.246·59-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 20.0T + 125T^{2} \) |
| 7 | \( 1 + 32.9T + 343T^{2} \) |
| 11 | \( 1 + 43.9T + 1.33e3T^{2} \) |
| 17 | \( 1 + 110.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 112.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 6.05T + 1.21e4T^{2} \) |
| 29 | \( 1 - 241.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 130.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 247.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 98.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 360.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 3.04T + 1.03e5T^{2} \) |
| 53 | \( 1 + 474.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 111.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 128.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 686.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 167.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 818.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 75.0T + 4.93e5T^{2} \) |
| 83 | \( 1 + 709.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 23.2T + 7.04e5T^{2} \) |
| 97 | \( 1 + 245.T + 9.12e5T^{2} \) |
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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
−8.324189168138424342983391985098, −7.47047405174546952696640587024, −6.89422284291272063654215849656, −6.19789191148801762566127514028, −5.02950192273716674734212841970, −4.31572256708616056605465654546, −3.32825564956196301927676815052, −2.76285385307629448194765221620, −0.66667639274466038704604853784, 0,
0.66667639274466038704604853784, 2.76285385307629448194765221620, 3.32825564956196301927676815052, 4.31572256708616056605465654546, 5.02950192273716674734212841970, 6.19789191148801762566127514028, 6.89422284291272063654215849656, 7.47047405174546952696640587024, 8.324189168138424342983391985098
