| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s + 3·11-s − 12-s + 4·13-s + 16-s + 18-s − 4·19-s + 3·22-s − 24-s + 4·26-s − 27-s + 9·29-s − 31-s + 32-s − 3·33-s + 36-s − 8·37-s − 4·38-s − 4·39-s + 10·43-s + 3·44-s + 6·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.904·11-s − 0.288·12-s + 1.10·13-s + 1/4·16-s + 0.235·18-s − 0.917·19-s + 0.639·22-s − 0.204·24-s + 0.784·26-s − 0.192·27-s + 1.67·29-s − 0.179·31-s + 0.176·32-s − 0.522·33-s + 1/6·36-s − 1.31·37-s − 0.648·38-s − 0.640·39-s + 1.52·43-s + 0.452·44-s + 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.099381945\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.099381945\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−7.77661403388345829925397928084, −6.87453886391034421744511633435, −6.42708985589432188060572692780, −5.87912503445314802197473662652, −5.09709067193903973081249507401, −4.23498041827927569450716824950, −3.83668560415003284884827901268, −2.82842476487235171282223948238, −1.76716327139387016284573306151, −0.862166222303041694164086476417,
0.862166222303041694164086476417, 1.76716327139387016284573306151, 2.82842476487235171282223948238, 3.83668560415003284884827901268, 4.23498041827927569450716824950, 5.09709067193903973081249507401, 5.87912503445314802197473662652, 6.42708985589432188060572692780, 6.87453886391034421744511633435, 7.77661403388345829925397928084
