| L(s) = 1 | + 2·7-s − 4·11-s − 4·13-s + 4·19-s + 2·23-s + 2·29-s − 4·37-s − 2·41-s + 6·43-s + 6·47-s − 3·49-s − 4·53-s − 12·59-s + 10·61-s − 14·67-s − 8·71-s + 8·73-s − 8·77-s + 16·79-s + 2·83-s − 6·89-s − 8·91-s + 16·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 1.20·11-s − 1.10·13-s + 0.917·19-s + 0.417·23-s + 0.371·29-s − 0.657·37-s − 0.312·41-s + 0.914·43-s + 0.875·47-s − 3/7·49-s − 0.549·53-s − 1.56·59-s + 1.28·61-s − 1.71·67-s − 0.949·71-s + 0.936·73-s − 0.911·77-s + 1.80·79-s + 0.219·83-s − 0.635·89-s − 0.838·91-s + 1.62·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
| 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
−16.37926961020176, −15.63981724246553, −15.45829378729099, −14.59260966320449, −14.28806080039422, −13.59322672486193, −13.09144867482140, −12.32755136039932, −12.00399546256356, −11.27549819754215, −10.65347245841540, −10.24881471812128, −9.512172106691461, −8.953070570484612, −8.174219361325910, −7.569844317841820, −7.347103862860115, −6.378148380067350, −5.563782847359060, −4.957627760717807, −4.670713661136509, −3.554999337526283, −2.775324181305250, −2.164835393483021, −1.150700887296982, 0,
1.150700887296982, 2.164835393483021, 2.775324181305250, 3.554999337526283, 4.670713661136509, 4.957627760717807, 5.563782847359060, 6.378148380067350, 7.347103862860115, 7.569844317841820, 8.174219361325910, 8.953070570484612, 9.512172106691461, 10.24881471812128, 10.65347245841540, 11.27549819754215, 12.00399546256356, 12.32755136039932, 13.09144867482140, 13.59322672486193, 14.28806080039422, 14.59260966320449, 15.45829378729099, 15.63981724246553, 16.37926961020176
