L(s) = 1 | − 2-s + 4-s + 7-s − 8-s + 4·11-s − 6·13-s − 14-s + 16-s + 4·17-s + 6·19-s − 4·22-s + 6·26-s + 28-s + 6·29-s − 4·31-s − 32-s − 4·34-s − 8·37-s − 6·38-s − 10·41-s + 2·43-s + 4·44-s + 10·47-s + 49-s − 6·52-s + 14·53-s − 56-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 1.20·11-s − 1.66·13-s − 0.267·14-s + 1/4·16-s + 0.970·17-s + 1.37·19-s − 0.852·22-s + 1.17·26-s + 0.188·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.685·34-s − 1.31·37-s − 0.973·38-s − 1.56·41-s + 0.304·43-s + 0.603·44-s + 1.45·47-s + 1/7·49-s − 0.832·52-s + 1.92·53-s − 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.457178903\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.457178903\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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
−8.802264981084487317388315950526, −7.88683732972392095219439624807, −7.24450885124220178032180679268, −6.76022217157049787656274957309, −5.57463696208936432634631731759, −5.00851543087317353748453641367, −3.85600654876603218806522778258, −2.94161321877212145155117878426, −1.85780267527966630511917932532, −0.838609342623186572423986224435,
0.838609342623186572423986224435, 1.85780267527966630511917932532, 2.94161321877212145155117878426, 3.85600654876603218806522778258, 5.00851543087317353748453641367, 5.57463696208936432634631731759, 6.76022217157049787656274957309, 7.24450885124220178032180679268, 7.88683732972392095219439624807, 8.802264981084487317388315950526