| L(s) = 1 | − 3-s + 2.54·5-s − 7-s + 9-s + 3.05·11-s + 13-s − 2.54·15-s + 1.34·17-s − 7.60·19-s + 21-s − 1.84·23-s + 1.50·25-s − 27-s − 5.60·29-s − 10.2·31-s − 3.05·33-s − 2.54·35-s − 9.20·37-s − 39-s − 8.04·41-s − 1.49·43-s + 2.54·45-s + 3.89·47-s + 49-s − 1.34·51-s + 0.502·53-s + 7.78·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.14·5-s − 0.377·7-s + 0.333·9-s + 0.920·11-s + 0.277·13-s − 0.658·15-s + 0.325·17-s − 1.74·19-s + 0.218·21-s − 0.384·23-s + 0.300·25-s − 0.192·27-s − 1.04·29-s − 1.84·31-s − 0.531·33-s − 0.431·35-s − 1.51·37-s − 0.160·39-s − 1.25·41-s − 0.228·43-s + 0.380·45-s + 0.567·47-s + 0.142·49-s − 0.187·51-s + 0.0690·53-s + 1.04·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 2.54T + 5T^{2} \) |
| 11 | \( 1 - 3.05T + 11T^{2} \) |
| 17 | \( 1 - 1.34T + 17T^{2} \) |
| 19 | \( 1 + 7.60T + 19T^{2} \) |
| 23 | \( 1 + 1.84T + 23T^{2} \) |
| 29 | \( 1 + 5.60T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + 9.20T + 37T^{2} \) |
| 41 | \( 1 + 8.04T + 41T^{2} \) |
| 43 | \( 1 + 1.49T + 43T^{2} \) |
| 47 | \( 1 - 3.89T + 47T^{2} \) |
| 53 | \( 1 - 0.502T + 53T^{2} \) |
| 59 | \( 1 - 4.28T + 59T^{2} \) |
| 61 | \( 1 + 0.683T + 61T^{2} \) |
| 67 | \( 1 - 7.68T + 67T^{2} \) |
| 71 | \( 1 - 14.2T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 + 4.91T + 79T^{2} \) |
| 83 | \( 1 - 1.20T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 + 7.18T + 97T^{2} \) |
| 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
−8.054411048763686913432713752507, −6.83536090003168028478131340648, −6.65157371843533836011966944169, −5.69706293111898123488290133480, −5.37770946520709482754223670889, −4.13896765872572997948131892815, −3.54612224333131570226586081691, −2.11990469519213814216392058765, −1.57616668352890074599738179797, 0,
1.57616668352890074599738179797, 2.11990469519213814216392058765, 3.54612224333131570226586081691, 4.13896765872572997948131892815, 5.37770946520709482754223670889, 5.69706293111898123488290133480, 6.65157371843533836011966944169, 6.83536090003168028478131340648, 8.054411048763686913432713752507
