| L(s) = 1 | + 2·7-s + 2·11-s − 2·17-s − 4·19-s − 2·29-s − 8·31-s + 4·37-s + 8·41-s + 8·43-s + 8·47-s − 3·49-s + 10·53-s − 6·59-s + 2·61-s + 12·67-s + 12·71-s + 2·73-s + 4·77-s − 8·79-s + 4·83-s + 12·89-s − 10·97-s − 10·101-s + 10·103-s + 12·107-s + 10·109-s − 6·113-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 0.603·11-s − 0.485·17-s − 0.917·19-s − 0.371·29-s − 1.43·31-s + 0.657·37-s + 1.24·41-s + 1.21·43-s + 1.16·47-s − 3/7·49-s + 1.37·53-s − 0.781·59-s + 0.256·61-s + 1.46·67-s + 1.42·71-s + 0.234·73-s + 0.455·77-s − 0.900·79-s + 0.439·83-s + 1.27·89-s − 1.01·97-s − 0.995·101-s + 0.985·103-s + 1.16·107-s + 0.957·109-s − 0.564·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.186264545\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.186264545\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.84220777970137204471300199726, −7.32480981864709847724946426972, −6.51405000412040067557498060587, −5.83541709553406979459162633957, −5.10185487915925817040243013651, −4.21518279984252753315744988478, −3.81472030343528485352654452433, −2.52335766056920657608934508094, −1.87616855604643254272182706140, −0.75633517224277193110679385394,
0.75633517224277193110679385394, 1.87616855604643254272182706140, 2.52335766056920657608934508094, 3.81472030343528485352654452433, 4.21518279984252753315744988478, 5.10185487915925817040243013651, 5.83541709553406979459162633957, 6.51405000412040067557498060587, 7.32480981864709847724946426972, 7.84220777970137204471300199726
