L(s) = 1 | − 5-s + 4·7-s − 3·9-s − 4·11-s + 2·17-s − 4·19-s + 4·23-s + 25-s − 2·29-s + 8·31-s − 4·35-s − 6·37-s + 6·41-s − 8·43-s + 3·45-s − 4·47-s + 9·49-s + 6·53-s + 4·55-s + 4·59-s − 2·61-s − 12·63-s − 8·67-s + 6·73-s − 16·77-s + 9·81-s + 16·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.51·7-s − 9-s − 1.20·11-s + 0.485·17-s − 0.917·19-s + 0.834·23-s + 1/5·25-s − 0.371·29-s + 1.43·31-s − 0.676·35-s − 0.986·37-s + 0.937·41-s − 1.21·43-s + 0.447·45-s − 0.583·47-s + 9/7·49-s + 0.824·53-s + 0.539·55-s + 0.520·59-s − 0.256·61-s − 1.51·63-s − 0.977·67-s + 0.702·73-s − 1.82·77-s + 81-s + 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.646807780\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.646807780\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.022217215535909925654587269096, −7.54659171755528277983966832262, −6.59612276407019711488721974723, −5.70144141475503043052967713028, −5.02286506551644332921966040958, −4.64266936056016518795462713453, −3.52853598753767170583266995525, −2.69438405187603052702862090211, −1.90270800756628574353833909448, −0.64728518773321187149973692487,
0.64728518773321187149973692487, 1.90270800756628574353833909448, 2.69438405187603052702862090211, 3.52853598753767170583266995525, 4.64266936056016518795462713453, 5.02286506551644332921966040958, 5.70144141475503043052967713028, 6.59612276407019711488721974723, 7.54659171755528277983966832262, 8.022217215535909925654587269096