| L(s) = 1 | + 2·3-s − 7-s + 9-s − 6·17-s − 6·19-s − 2·21-s − 23-s − 5·25-s − 4·27-s − 6·29-s − 8·31-s + 2·37-s − 2·41-s + 8·43-s + 8·47-s + 49-s − 12·51-s − 2·53-s − 12·57-s + 6·59-s − 63-s + 12·67-s − 2·69-s + 8·71-s − 6·73-s − 10·75-s + 16·79-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.377·7-s + 1/3·9-s − 1.45·17-s − 1.37·19-s − 0.436·21-s − 0.208·23-s − 25-s − 0.769·27-s − 1.11·29-s − 1.43·31-s + 0.328·37-s − 0.312·41-s + 1.21·43-s + 1.16·47-s + 1/7·49-s − 1.68·51-s − 0.274·53-s − 1.58·57-s + 0.781·59-s − 0.125·63-s + 1.46·67-s − 0.240·69-s + 0.949·71-s − 0.702·73-s − 1.15·75-s + 1.80·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.593920831656403222628481743690, −7.87315473667389214474206232917, −7.10809146207233898224339121634, −6.27617220116716397525744333594, −5.43449215492340566701756875947, −4.10672982710302442963515478764, −3.76350343323065855318078905888, −2.46956130283940301981085030530, −2.01617019085807731991022185363, 0,
2.01617019085807731991022185363, 2.46956130283940301981085030530, 3.76350343323065855318078905888, 4.10672982710302442963515478764, 5.43449215492340566701756875947, 6.27617220116716397525744333594, 7.10809146207233898224339121634, 7.87315473667389214474206232917, 8.593920831656403222628481743690
