| L(s) = 1 | + 7-s + 4·11-s + 13-s + 4·17-s − 19-s + 4·23-s − 4·29-s + 5·31-s + 6·37-s − 12·41-s + 5·43-s − 8·47-s − 6·49-s − 12·53-s + 8·59-s + 7·61-s + 13·67-s + 12·71-s + 6·73-s + 4·77-s − 12·79-s + 8·83-s + 91-s + 13·97-s − 12·101-s − 4·103-s − 12·107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 1.20·11-s + 0.277·13-s + 0.970·17-s − 0.229·19-s + 0.834·23-s − 0.742·29-s + 0.898·31-s + 0.986·37-s − 1.87·41-s + 0.762·43-s − 1.16·47-s − 6/7·49-s − 1.64·53-s + 1.04·59-s + 0.896·61-s + 1.58·67-s + 1.42·71-s + 0.702·73-s + 0.455·77-s − 1.35·79-s + 0.878·83-s + 0.104·91-s + 1.31·97-s − 1.19·101-s − 0.394·103-s − 1.16·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.300437062\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.300437062\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−8.361945858397800869766843110667, −8.032572103090759902035951839090, −6.90954094133465220310457882169, −6.47268772919931201622475987172, −5.50347388995332359515940196019, −4.76769755972939386850880132916, −3.84799867892829775034307698073, −3.13800853113216742370456442221, −1.85990902738589702547809446026, −0.953561415165416515731374030537,
0.953561415165416515731374030537, 1.85990902738589702547809446026, 3.13800853113216742370456442221, 3.84799867892829775034307698073, 4.76769755972939386850880132916, 5.50347388995332359515940196019, 6.47268772919931201622475987172, 6.90954094133465220310457882169, 8.032572103090759902035951839090, 8.361945858397800869766843110667
