L(s) = 1 | + 5-s − 7-s + 6·11-s − 13-s − 19-s + 6·23-s + 25-s + 6·29-s + 8·31-s − 35-s − 7·37-s − 6·41-s − 4·43-s + 12·47-s − 6·49-s − 6·53-s + 6·55-s + 11·61-s − 65-s − 7·67-s − 6·71-s + 11·73-s − 6·77-s − 79-s + 6·83-s − 12·89-s + 91-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.377·7-s + 1.80·11-s − 0.277·13-s − 0.229·19-s + 1.25·23-s + 1/5·25-s + 1.11·29-s + 1.43·31-s − 0.169·35-s − 1.15·37-s − 0.937·41-s − 0.609·43-s + 1.75·47-s − 6/7·49-s − 0.824·53-s + 0.809·55-s + 1.40·61-s − 0.124·65-s − 0.855·67-s − 0.712·71-s + 1.28·73-s − 0.683·77-s − 0.112·79-s + 0.658·83-s − 1.27·89-s + 0.104·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.638542216\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.638542216\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−10.74794183187189613430165450110, −9.844564467715010164242704464375, −9.127418035079580641451740646574, −8.343543745788865357315277029134, −6.81615637958252312767893811403, −6.53488198804866315995421780383, −5.20522128847215086701609074104, −4.09617400536347052792491503589, −2.89403046630752699619618864147, −1.31220016809550134539290495913,
1.31220016809550134539290495913, 2.89403046630752699619618864147, 4.09617400536347052792491503589, 5.20522128847215086701609074104, 6.53488198804866315995421780383, 6.81615637958252312767893811403, 8.343543745788865357315277029134, 9.127418035079580641451740646574, 9.844564467715010164242704464375, 10.74794183187189613430165450110