| L(s) = 1 | + 3-s − 4.32·7-s + 9-s + 0.584·11-s − 0.966·13-s + 2.32·17-s − 5.37·19-s − 4.32·21-s + 1.35·23-s + 27-s − 0.706·29-s + 8.48·31-s + 0.584·33-s + 6.11·37-s − 0.966·39-s + 11.0·41-s + 7.03·43-s − 9.06·47-s + 11.7·49-s + 2.32·51-s − 8.90·53-s − 5.37·57-s − 7.29·59-s − 4.81·61-s − 4.32·63-s + 0.376·67-s + 1.35·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.63·7-s + 0.333·9-s + 0.176·11-s − 0.267·13-s + 0.563·17-s − 1.23·19-s − 0.943·21-s + 0.283·23-s + 0.192·27-s − 0.131·29-s + 1.52·31-s + 0.101·33-s + 1.00·37-s − 0.154·39-s + 1.72·41-s + 1.07·43-s − 1.32·47-s + 1.67·49-s + 0.325·51-s − 1.22·53-s − 0.712·57-s − 0.949·59-s − 0.616·61-s − 0.544·63-s + 0.0459·67-s + 0.163·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 4.32T + 7T^{2} \) |
| 11 | \( 1 - 0.584T + 11T^{2} \) |
| 13 | \( 1 + 0.966T + 13T^{2} \) |
| 17 | \( 1 - 2.32T + 17T^{2} \) |
| 19 | \( 1 + 5.37T + 19T^{2} \) |
| 23 | \( 1 - 1.35T + 23T^{2} \) |
| 29 | \( 1 + 0.706T + 29T^{2} \) |
| 31 | \( 1 - 8.48T + 31T^{2} \) |
| 37 | \( 1 - 6.11T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 - 7.03T + 43T^{2} \) |
| 47 | \( 1 + 9.06T + 47T^{2} \) |
| 53 | \( 1 + 8.90T + 53T^{2} \) |
| 59 | \( 1 + 7.29T + 59T^{2} \) |
| 61 | \( 1 + 4.81T + 61T^{2} \) |
| 67 | \( 1 - 0.376T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 + 0.213T + 73T^{2} \) |
| 79 | \( 1 + 7.02T + 79T^{2} \) |
| 83 | \( 1 + 16.1T + 83T^{2} \) |
| 89 | \( 1 + 14.0T + 89T^{2} \) |
| 97 | \( 1 + 15.7T + 97T^{2} \) |
| 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
−7.65556498940680176622933398980, −6.63538121741455798042112707350, −6.40716096702515458295878736288, −5.58563310152273468471065484841, −4.42946335116055519192005786328, −3.94701492282413512171332066806, −2.87802455026655586500457441964, −2.66404215141461288589152969237, −1.25813089195527744203788111994, 0,
1.25813089195527744203788111994, 2.66404215141461288589152969237, 2.87802455026655586500457441964, 3.94701492282413512171332066806, 4.42946335116055519192005786328, 5.58563310152273468471065484841, 6.40716096702515458295878736288, 6.63538121741455798042112707350, 7.65556498940680176622933398980
