L(s) = 1 | − 1.73i·7-s + (−0.5 + 0.866i)13-s + 19-s + (0.5 − 0.866i)25-s − 1.73i·31-s − 37-s + (1.5 − 0.866i)43-s − 1.99·49-s + (0.5 − 0.866i)61-s + (−1.5 − 0.866i)67-s + (−0.5 − 0.866i)73-s + (−1.5 + 0.866i)79-s + (1.49 + 0.866i)91-s + (1 + 1.73i)97-s − 1.73i·103-s + ⋯ |
L(s) = 1 | − 1.73i·7-s + (−0.5 + 0.866i)13-s + 19-s + (0.5 − 0.866i)25-s − 1.73i·31-s − 37-s + (1.5 − 0.866i)43-s − 1.99·49-s + (0.5 − 0.866i)61-s + (−1.5 − 0.866i)67-s + (−0.5 − 0.866i)73-s + (−1.5 + 0.866i)79-s + (1.49 + 0.866i)91-s + (1 + 1.73i)97-s − 1.73i·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.138800854\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.138800854\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + 1.73iT - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + 1.73iT - T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−8.943994934304339261617236592804, −7.87273262613030846417016401448, −7.34073364062009789488295651085, −6.77888406630811769064961417941, −5.83912396319886297117849941603, −4.70815618028868880075134397474, −4.17810278864933120319254758079, −3.31808992649296495843495233733, −2.05880563793082294983593070623, −0.76959534355878156740201573646,
1.50606362395437365325097187522, 2.76213216514709892984274457454, 3.18985694617474133857687924156, 4.66579629539841551547516813598, 5.46945916640562172902027589448, 5.79609977516594648501204724341, 6.96516226400799493069668125762, 7.63823020565407308784191310789, 8.655110398597480460679181888037, 8.952910772874176106280888074193