| L(s) = 1 | − 2·3-s − 2·4-s + 4·7-s + 9-s + 11-s + 4·12-s − 5·13-s + 4·16-s − 7·17-s − 8·21-s + 4·27-s − 8·28-s + 8·29-s + 4·31-s − 2·33-s − 2·36-s − 10·37-s + 10·39-s + 2·41-s − 43-s − 2·44-s + 7·47-s − 8·48-s + 9·49-s + 14·51-s + 10·52-s + 10·53-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 4-s + 1.51·7-s + 1/3·9-s + 0.301·11-s + 1.15·12-s − 1.38·13-s + 16-s − 1.69·17-s − 1.74·21-s + 0.769·27-s − 1.51·28-s + 1.48·29-s + 0.718·31-s − 0.348·33-s − 1/3·36-s − 1.64·37-s + 1.60·39-s + 0.312·41-s − 0.152·43-s − 0.301·44-s + 1.02·47-s − 1.15·48-s + 9/7·49-s + 1.96·51-s + 1.38·52-s + 1.37·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7793671237\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7793671237\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−10.11669421323854871335783858435, −8.873579525530716141577193189101, −8.441989212576646095053257580794, −7.33679024074670642761924458685, −6.43229114637235675401110555713, −5.19379890049721665286512086868, −4.91792719629567512607695747403, −4.15420842176846791740104754810, −2.29094053936300425209093701972, −0.72089895473037622488997239722,
0.72089895473037622488997239722, 2.29094053936300425209093701972, 4.15420842176846791740104754810, 4.91792719629567512607695747403, 5.19379890049721665286512086868, 6.43229114637235675401110555713, 7.33679024074670642761924458685, 8.441989212576646095053257580794, 8.873579525530716141577193189101, 10.11669421323854871335783858435
