| L(s) = 1 | + 3-s + 5-s + 2·7-s + 9-s − 2·11-s + 15-s − 7·17-s − 6·19-s + 2·21-s + 6·23-s − 4·25-s + 27-s − 29-s + 4·31-s − 2·33-s + 2·35-s − 37-s − 9·41-s − 6·43-s + 45-s + 6·47-s − 3·49-s − 7·51-s − 9·53-s − 2·55-s − 6·57-s + 61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.603·11-s + 0.258·15-s − 1.69·17-s − 1.37·19-s + 0.436·21-s + 1.25·23-s − 4/5·25-s + 0.192·27-s − 0.185·29-s + 0.718·31-s − 0.348·33-s + 0.338·35-s − 0.164·37-s − 1.40·41-s − 0.914·43-s + 0.149·45-s + 0.875·47-s − 3/7·49-s − 0.980·51-s − 1.23·53-s − 0.269·55-s − 0.794·57-s + 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.57031412080773567124460337556, −6.73695508763117480786565549235, −6.27985727107805508897931687097, −5.18643923324411764357837439133, −4.69925490877292098043060131009, −3.98003983600478068107013638489, −2.91755777743331542964048204204, −2.18769823435216498380482100613, −1.57317668664651057664712541934, 0,
1.57317668664651057664712541934, 2.18769823435216498380482100613, 2.91755777743331542964048204204, 3.98003983600478068107013638489, 4.69925490877292098043060131009, 5.18643923324411764357837439133, 6.27985727107805508897931687097, 6.73695508763117480786565549235, 7.57031412080773567124460337556
