| L(s) = 1 | + 2-s − 3-s − 4-s − 6-s + 4·7-s − 3·8-s + 9-s + 12-s + 2·13-s + 4·14-s − 16-s + 2·17-s + 18-s + 8·19-s − 4·21-s + 4·23-s + 3·24-s + 2·26-s − 27-s − 4·28-s + 4·29-s − 8·31-s + 5·32-s + 2·34-s − 36-s + 8·37-s + 8·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s + 1.51·7-s − 1.06·8-s + 1/3·9-s + 0.288·12-s + 0.554·13-s + 1.06·14-s − 1/4·16-s + 0.485·17-s + 0.235·18-s + 1.83·19-s − 0.872·21-s + 0.834·23-s + 0.612·24-s + 0.392·26-s − 0.192·27-s − 0.755·28-s + 0.742·29-s − 1.43·31-s + 0.883·32-s + 0.342·34-s − 1/6·36-s + 1.31·37-s + 1.29·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.808530482\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.808530482\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 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 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.77265486611979398528312164208, −7.03565448696680126422915412250, −5.98531246931203881217695382070, −5.56994938875895586898916771178, −4.88444892499607193134431413471, −4.52861632502101416499953505621, −3.59493184135857138797421970462, −2.87903477757775277290344936537, −1.53402504282732439976312018354, −0.835398163127846365332016843080,
0.835398163127846365332016843080, 1.53402504282732439976312018354, 2.87903477757775277290344936537, 3.59493184135857138797421970462, 4.52861632502101416499953505621, 4.88444892499607193134431413471, 5.56994938875895586898916771178, 5.98531246931203881217695382070, 7.03565448696680126422915412250, 7.77265486611979398528312164208
