| L(s) = 1 | + 3-s + 9-s + 11-s + 4·13-s − 2·17-s + 4·19-s − 4·23-s + 27-s + 8·29-s − 8·31-s + 33-s + 6·37-s + 4·39-s + 6·41-s + 8·43-s + 4·47-s − 2·51-s + 12·53-s + 4·57-s − 6·59-s + 2·61-s − 2·67-s − 4·69-s − 4·73-s + 2·79-s + 81-s − 10·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.301·11-s + 1.10·13-s − 0.485·17-s + 0.917·19-s − 0.834·23-s + 0.192·27-s + 1.48·29-s − 1.43·31-s + 0.174·33-s + 0.986·37-s + 0.640·39-s + 0.937·41-s + 1.21·43-s + 0.583·47-s − 0.280·51-s + 1.64·53-s + 0.529·57-s − 0.781·59-s + 0.256·61-s − 0.244·67-s − 0.481·69-s − 0.468·73-s + 0.225·79-s + 1/9·81-s − 1.09·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.938633180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.938633180\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 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 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−12.70981231962879, −12.13736719636663, −11.74286159147456, −11.23686996693704, −10.79294413839587, −10.33797542340622, −9.875926471863613, −9.197560531930471, −9.068771719862088, −8.512977821735216, −8.077922137282263, −7.434767328599884, −7.264919898685980, −6.519257692702883, −5.989063749025917, −5.757666834600202, −5.007590396962103, −4.292626156637806, −4.085412691308252, −3.467560075112341, −2.939214707260428, −2.354872684348635, −1.809935045034708, −1.057749816188947, −0.6381589673512795,
0.6381589673512795, 1.057749816188947, 1.809935045034708, 2.354872684348635, 2.939214707260428, 3.467560075112341, 4.085412691308252, 4.292626156637806, 5.007590396962103, 5.757666834600202, 5.989063749025917, 6.519257692702883, 7.264919898685980, 7.434767328599884, 8.077922137282263, 8.512977821735216, 9.068771719862088, 9.197560531930471, 9.875926471863613, 10.33797542340622, 10.79294413839587, 11.23686996693704, 11.74286159147456, 12.13736719636663, 12.70981231962879
