| L(s) = 1 | + 2·5-s + 7-s + 6·17-s + 4·19-s − 25-s + 2·29-s + 8·31-s + 2·35-s − 6·37-s + 6·41-s + 4·43-s − 12·47-s + 49-s + 2·53-s − 2·61-s + 12·67-s + 4·71-s − 10·73-s − 16·79-s + 12·85-s − 2·89-s + 8·95-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.377·7-s + 1.45·17-s + 0.917·19-s − 1/5·25-s + 0.371·29-s + 1.43·31-s + 0.338·35-s − 0.986·37-s + 0.937·41-s + 0.609·43-s − 1.75·47-s + 1/7·49-s + 0.274·53-s − 0.256·61-s + 1.46·67-s + 0.474·71-s − 1.17·73-s − 1.80·79-s + 1.30·85-s − 0.211·89-s + 0.820·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.035489850\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.035489850\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 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 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.00464123670189, −13.49210024646118, −13.03319019539336, −12.35246913556506, −12.03037350986169, −11.45871802223532, −11.00384991030620, −10.18281793095504, −9.964622356044448, −9.632827017909104, −8.913068450447088, −8.342714826302650, −7.892509269992149, −7.322274885780135, −6.795310063754959, −5.975596907366501, −5.809734895546520, −5.104536584228383, −4.692296084396337, −3.885560722216249, −3.192898544717624, −2.727529480470073, −1.906153997664113, −1.327212088072859, −0.6865725812934510,
0.6865725812934510, 1.327212088072859, 1.906153997664113, 2.727529480470073, 3.192898544717624, 3.885560722216249, 4.692296084396337, 5.104536584228383, 5.809734895546520, 5.975596907366501, 6.795310063754959, 7.322274885780135, 7.892509269992149, 8.342714826302650, 8.913068450447088, 9.632827017909104, 9.964622356044448, 10.18281793095504, 11.00384991030620, 11.45871802223532, 12.03037350986169, 12.35246913556506, 13.03319019539336, 13.49210024646118, 14.00464123670189
