| L(s) = 1 | − 4·7-s − 4·11-s + 2·13-s + 2·17-s − 19-s + 4·23-s − 6·29-s + 2·37-s + 6·41-s + 12·43-s + 4·47-s + 9·49-s − 2·53-s + 12·59-s + 10·61-s + 4·67-s − 10·73-s + 16·77-s + 8·79-s − 12·83-s + 14·89-s − 8·91-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 1.20·11-s + 0.554·13-s + 0.485·17-s − 0.229·19-s + 0.834·23-s − 1.11·29-s + 0.328·37-s + 0.937·41-s + 1.82·43-s + 0.583·47-s + 9/7·49-s − 0.274·53-s + 1.56·59-s + 1.28·61-s + 0.488·67-s − 1.17·73-s + 1.82·77-s + 0.900·79-s − 1.31·83-s + 1.48·89-s − 0.838·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.133943246\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.133943246\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−12.94524133842415, −12.42742225610591, −11.99346858060326, −11.18966125720292, −10.95311392101923, −10.49045573967044, −9.940294752880482, −9.590994337428258, −9.130060530534116, −8.675437651869254, −8.068675721049073, −7.580813303719281, −7.108125170024634, −6.719147867480564, −6.028022297671916, −5.623773454889960, −5.405354382812166, −4.485311686902240, −4.017680132437176, −3.458578956911894, −2.952515935504809, −2.521530106859499, −1.908730745679543, −0.8206419096845997, −0.5218940450563302,
0.5218940450563302, 0.8206419096845997, 1.908730745679543, 2.521530106859499, 2.952515935504809, 3.458578956911894, 4.017680132437176, 4.485311686902240, 5.405354382812166, 5.623773454889960, 6.028022297671916, 6.719147867480564, 7.108125170024634, 7.580813303719281, 8.068675721049073, 8.675437651869254, 9.130060530534116, 9.590994337428258, 9.940294752880482, 10.49045573967044, 10.95311392101923, 11.18966125720292, 11.99346858060326, 12.42742225610591, 12.94524133842415
