L(s) = 1 | + 1.41·5-s − 1.41i·17-s + 1.00·25-s + 1.41·29-s + 2i·37-s − 1.41i·41-s − 49-s − 1.41·53-s + 2i·61-s − 2.00i·85-s − 1.41i·89-s + 1.41·101-s + 1.41i·113-s + ⋯ |
L(s) = 1 | + 1.41·5-s − 1.41i·17-s + 1.00·25-s + 1.41·29-s + 2i·37-s − 1.41i·41-s − 49-s − 1.41·53-s + 2i·61-s − 2.00i·85-s − 1.41i·89-s + 1.41·101-s + 1.41i·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.540766485\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.540766485\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 1.41T + T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + 1.41iT - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - 1.41T + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - 2iT - T^{2} \) |
| 41 | \( 1 + 1.41iT - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.41T + T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - 2iT - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + 1.41iT - T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.212266051383144554642996347978, −8.597058472575831558684751570405, −7.56946764151252692614723732508, −6.69382739238078947840478299283, −6.11564829054366919430590158364, −5.19513091173301948970831833379, −4.62177332894830209763602781815, −3.16198802717098525385711424677, −2.41435388271804923174775327567, −1.28015385204437275932220396026,
1.45731166314950487121510487204, 2.29517728673042659054256203652, 3.37968741405111118571776920880, 4.51291407764597439002672536385, 5.37322327612649580110253446526, 6.21368904902907447871323542327, 6.55474490141589343664411811676, 7.80007317043602583412837661978, 8.476116353491173746067148285916, 9.374561146228005112665045471263