| L(s) = 1 | + 3-s − 3·7-s + 9-s + 2·11-s + 13-s − 2·17-s − 5·19-s − 3·21-s + 6·23-s + 27-s − 10·29-s + 3·31-s + 2·33-s + 2·37-s + 39-s − 8·41-s − 43-s + 2·47-s + 2·49-s − 2·51-s − 4·53-s − 5·57-s − 10·59-s − 7·61-s − 3·63-s + 3·67-s + 6·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.13·7-s + 1/3·9-s + 0.603·11-s + 0.277·13-s − 0.485·17-s − 1.14·19-s − 0.654·21-s + 1.25·23-s + 0.192·27-s − 1.85·29-s + 0.538·31-s + 0.348·33-s + 0.328·37-s + 0.160·39-s − 1.24·41-s − 0.152·43-s + 0.291·47-s + 2/7·49-s − 0.280·51-s − 0.549·53-s − 0.662·57-s − 1.30·59-s − 0.896·61-s − 0.377·63-s + 0.366·67-s + 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−8.040086974208892692137412150811, −7.01259548848434300802989966251, −6.63552338173317415872093243656, −5.90404919386938898421911646081, −4.85692283621813202383375265351, −3.96102438209774458569462189574, −3.36957507180644347787302261531, −2.51201353173268436104607760745, −1.48067503408916751090348731726, 0,
1.48067503408916751090348731726, 2.51201353173268436104607760745, 3.36957507180644347787302261531, 3.96102438209774458569462189574, 4.85692283621813202383375265351, 5.90404919386938898421911646081, 6.63552338173317415872093243656, 7.01259548848434300802989966251, 8.040086974208892692137412150811
