L(s) = 1 | − 3-s − 4·7-s + 9-s + 2·13-s + 6·17-s + 4·21-s − 4·23-s − 27-s − 2·29-s + 8·31-s − 6·37-s − 2·39-s − 6·41-s + 12·43-s − 12·47-s + 9·49-s − 6·51-s + 10·53-s − 8·59-s − 10·61-s − 4·63-s − 12·67-s + 4·69-s − 8·71-s − 10·73-s − 16·79-s + 81-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s + 0.554·13-s + 1.45·17-s + 0.872·21-s − 0.834·23-s − 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.986·37-s − 0.320·39-s − 0.937·41-s + 1.82·43-s − 1.75·47-s + 9/7·49-s − 0.840·51-s + 1.37·53-s − 1.04·59-s − 1.28·61-s − 0.503·63-s − 1.46·67-s + 0.481·69-s − 0.949·71-s − 1.17·73-s − 1.80·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−8.644654543454582875017439509956, −7.69656249424289905776777126602, −6.93787727800819058796608060670, −6.05737762363916488521953017065, −5.78024039977257919368245069301, −4.56095924312758711126735516610, −3.58644340419390986422688793230, −2.90871427284981407321742713309, −1.35684162372205646349678849230, 0,
1.35684162372205646349678849230, 2.90871427284981407321742713309, 3.58644340419390986422688793230, 4.56095924312758711126735516610, 5.78024039977257919368245069301, 6.05737762363916488521953017065, 6.93787727800819058796608060670, 7.69656249424289905776777126602, 8.644654543454582875017439509956