L(s) = 1 | − 3·3-s − 13·7-s + 9·9-s − 6·11-s − 5·13-s + 78·17-s − 65·19-s + 39·21-s + 138·23-s − 27·27-s + 66·29-s − 299·31-s + 18·33-s + 214·37-s + 15·39-s + 360·41-s + 203·43-s + 78·47-s − 174·49-s − 234·51-s − 636·53-s + 195·57-s − 786·59-s + 467·61-s − 117·63-s − 217·67-s − 414·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.701·7-s + 1/3·9-s − 0.164·11-s − 0.106·13-s + 1.11·17-s − 0.784·19-s + 0.405·21-s + 1.25·23-s − 0.192·27-s + 0.422·29-s − 1.73·31-s + 0.0949·33-s + 0.950·37-s + 0.0615·39-s + 1.37·41-s + 0.719·43-s + 0.242·47-s − 0.507·49-s − 0.642·51-s − 1.64·53-s + 0.453·57-s − 1.73·59-s + 0.980·61-s − 0.233·63-s − 0.395·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 13 T + p^{3} T^{2} \) |
| 11 | \( 1 + 6 T + p^{3} T^{2} \) |
| 13 | \( 1 + 5 T + p^{3} T^{2} \) |
| 17 | \( 1 - 78 T + p^{3} T^{2} \) |
| 19 | \( 1 + 65 T + p^{3} T^{2} \) |
| 23 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 29 | \( 1 - 66 T + p^{3} T^{2} \) |
| 31 | \( 1 + 299 T + p^{3} T^{2} \) |
| 37 | \( 1 - 214 T + p^{3} T^{2} \) |
| 41 | \( 1 - 360 T + p^{3} T^{2} \) |
| 43 | \( 1 - 203 T + p^{3} T^{2} \) |
| 47 | \( 1 - 78 T + p^{3} T^{2} \) |
| 53 | \( 1 + 12 p T + p^{3} T^{2} \) |
| 59 | \( 1 + 786 T + p^{3} T^{2} \) |
| 61 | \( 1 - 467 T + p^{3} T^{2} \) |
| 67 | \( 1 + 217 T + p^{3} T^{2} \) |
| 71 | \( 1 - 360 T + p^{3} T^{2} \) |
| 73 | \( 1 - 286 T + p^{3} T^{2} \) |
| 79 | \( 1 + 272 T + p^{3} T^{2} \) |
| 83 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 89 | \( 1 + p^{3} T^{2} \) |
| 97 | \( 1 - 511 T + p^{3} T^{2} \) |
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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
−9.233215057113962041448410594792, −8.019814081191554426939270477612, −7.25828701611279480605294335646, −6.37525160349942488595371699502, −5.65859881209199010729043775493, −4.74067969700779109860110539372, −3.67031190205928047663633085685, −2.66232830582804892453275086523, −1.20735769226286796617619360174, 0,
1.20735769226286796617619360174, 2.66232830582804892453275086523, 3.67031190205928047663633085685, 4.74067969700779109860110539372, 5.65859881209199010729043775493, 6.37525160349942488595371699502, 7.25828701611279480605294335646, 8.019814081191554426939270477612, 9.233215057113962041448410594792