L(s) = 1 | − 2·5-s − 7-s − 3·9-s + 4·11-s + 4·13-s − 8·17-s + 2·19-s − 23-s − 25-s + 2·29-s + 6·31-s + 2·35-s − 10·37-s + 6·41-s + 8·43-s + 6·45-s − 6·47-s + 49-s + 2·53-s − 8·55-s + 10·61-s + 3·63-s − 8·65-s − 8·67-s + 12·71-s + 6·73-s − 4·77-s + ⋯ |
L(s) = 1 | − 0.894·5-s − 0.377·7-s − 9-s + 1.20·11-s + 1.10·13-s − 1.94·17-s + 0.458·19-s − 0.208·23-s − 1/5·25-s + 0.371·29-s + 1.07·31-s + 0.338·35-s − 1.64·37-s + 0.937·41-s + 1.21·43-s + 0.894·45-s − 0.875·47-s + 1/7·49-s + 0.274·53-s − 1.07·55-s + 1.28·61-s + 0.377·63-s − 0.992·65-s − 0.977·67-s + 1.42·71-s + 0.702·73-s − 0.455·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.215526392\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.215526392\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−8.753366855177622331907936695054, −8.368735013366825232532695466513, −7.33044829921278694710892432356, −6.46272750767080528681209373036, −6.06938450181969946339937340496, −4.80873033261637061271558405047, −3.94582309832033614475384163494, −3.37902247931740528885602458644, −2.18320730598253126880853975506, −0.68306258737287573659672112948,
0.68306258737287573659672112948, 2.18320730598253126880853975506, 3.37902247931740528885602458644, 3.94582309832033614475384163494, 4.80873033261637061271558405047, 6.06938450181969946339937340496, 6.46272750767080528681209373036, 7.33044829921278694710892432356, 8.368735013366825232532695466513, 8.753366855177622331907936695054