L(s) = 1 | + i·3-s + 2·9-s − 11-s − 5i·13-s − i·17-s − 6·19-s + 4i·23-s + 5i·27-s − 3·29-s − 2·31-s − i·33-s + 8i·37-s + 5·39-s + 10·41-s + 2i·43-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 0.666·9-s − 0.301·11-s − 1.38i·13-s − 0.242i·17-s − 1.37·19-s + 0.834i·23-s + 0.962i·27-s − 0.557·29-s − 0.359·31-s − 0.174i·33-s + 1.31i·37-s + 0.800·39-s + 1.56·41-s + 0.304i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.747843881\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.747843881\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - iT - 3T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 + 5iT - 13T^{2} \) |
| 17 | \( 1 + iT - 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 - 7iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 - 14T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 - 14iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 - 11T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 - 4T + 89T^{2} \) |
| 97 | \( 1 - 3iT - 97T^{2} \) |
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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
−8.336770698229542027837911920382, −7.71655128925852666411015085302, −7.04511910512541226113017577682, −6.09997131661849737338739933214, −5.41778465838601779723156813228, −4.67055415451555320605003897895, −3.92929004918294681349683772856, −3.13606487592853785355815178672, −2.16764535844585254365487144201, −0.915198530301956344907416148185,
0.57568665876619869641404572650, 1.97484523556584736182611834869, 2.26362799469921881336522685971, 3.86394934816305731368724420547, 4.22957254357812606690900974505, 5.21527176738940598185486369507, 6.18817676576575173494178406956, 6.74502990089264962906491116674, 7.29771310427847617219442909229, 8.094554842298300230702934709854