| L(s) = 1 | − 3·3-s − 5.89·5-s − 4.42·7-s + 9·9-s − 68.2·11-s + 17.6·15-s + 1.73·17-s + 148.·19-s + 13.2·21-s + 152.·23-s − 90.2·25-s − 27·27-s − 298.·29-s + 184.·31-s + 204.·33-s + 26.0·35-s + 220.·37-s + 48.0·41-s + 193.·43-s − 53.0·45-s + 276.·47-s − 323.·49-s − 5.19·51-s + 513.·53-s + 402.·55-s − 444.·57-s − 507.·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.527·5-s − 0.238·7-s + 0.333·9-s − 1.87·11-s + 0.304·15-s + 0.0246·17-s + 1.78·19-s + 0.137·21-s + 1.37·23-s − 0.721·25-s − 0.192·27-s − 1.91·29-s + 1.06·31-s + 1.08·33-s + 0.125·35-s + 0.979·37-s + 0.183·41-s + 0.686·43-s − 0.175·45-s + 0.856·47-s − 0.942·49-s − 0.0142·51-s + 1.33·53-s + 0.986·55-s − 1.03·57-s − 1.12·59-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 | \( 1 + 5.89T + 125T^{2} \) |
| 7 | \( 1 + 4.42T + 343T^{2} \) |
| 11 | \( 1 + 68.2T + 1.33e3T^{2} \) |
| 17 | \( 1 - 1.73T + 4.91e3T^{2} \) |
| 19 | \( 1 - 148.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 152.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 298.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 184.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 220.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 48.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 193.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 276.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 513.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 507.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 218.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 974.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 843.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 821.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.16e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 19.0T + 5.71e5T^{2} \) |
| 89 | \( 1 + 406.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.12e3T + 9.12e5T^{2} \) |
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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
−8.159902166653312788707628720722, −7.56473624361092276886248318910, −7.04171272373373826003284061019, −5.70335515903627983023681512959, −5.37964822971078201063341730482, −4.41231214638140460012715535426, −3.32492846309487818196390664500, −2.50111624874819794620637804731, −0.987040604113031043724576198276, 0,
0.987040604113031043724576198276, 2.50111624874819794620637804731, 3.32492846309487818196390664500, 4.41231214638140460012715535426, 5.37964822971078201063341730482, 5.70335515903627983023681512959, 7.04171272373373826003284061019, 7.56473624361092276886248318910, 8.159902166653312788707628720722
