| L(s) = 1 | + 2.67·3-s + 1.14·7-s + 4.14·9-s + 3.81·11-s − 13-s − 7.34·17-s + 5.52·19-s + 3.05·21-s + 6.67·23-s + 3.05·27-s + 2.85·29-s − 0.183·31-s + 10.2·33-s − 6.48·37-s − 2.67·39-s − 7.34·41-s − 2.67·43-s + 2.85·47-s − 5.69·49-s − 19.6·51-s + 13.6·53-s + 14.7·57-s + 5.16·59-s + 5.14·61-s + 4.74·63-s + 2.48·67-s + 17.8·69-s + ⋯ |
| L(s) = 1 | + 1.54·3-s + 0.432·7-s + 1.38·9-s + 1.15·11-s − 0.277·13-s − 1.78·17-s + 1.26·19-s + 0.667·21-s + 1.39·23-s + 0.588·27-s + 0.530·29-s − 0.0329·31-s + 1.77·33-s − 1.06·37-s − 0.427·39-s − 1.14·41-s − 0.407·43-s + 0.416·47-s − 0.813·49-s − 2.74·51-s + 1.87·53-s + 1.95·57-s + 0.672·59-s + 0.658·61-s + 0.597·63-s + 0.304·67-s + 2.14·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.109789955\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.109789955\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 2.67T + 3T^{2} \) |
| 7 | \( 1 - 1.14T + 7T^{2} \) |
| 11 | \( 1 - 3.81T + 11T^{2} \) |
| 17 | \( 1 + 7.34T + 17T^{2} \) |
| 19 | \( 1 - 5.52T + 19T^{2} \) |
| 23 | \( 1 - 6.67T + 23T^{2} \) |
| 29 | \( 1 - 2.85T + 29T^{2} \) |
| 31 | \( 1 + 0.183T + 31T^{2} \) |
| 37 | \( 1 + 6.48T + 37T^{2} \) |
| 41 | \( 1 + 7.34T + 41T^{2} \) |
| 43 | \( 1 + 2.67T + 43T^{2} \) |
| 47 | \( 1 - 2.85T + 47T^{2} \) |
| 53 | \( 1 - 13.6T + 53T^{2} \) |
| 59 | \( 1 - 5.16T + 59T^{2} \) |
| 61 | \( 1 - 5.14T + 61T^{2} \) |
| 67 | \( 1 - 2.48T + 67T^{2} \) |
| 71 | \( 1 - 3.81T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 - 9.34T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 + 7.71T + 97T^{2} \) |
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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
−9.359627193875159814630097624569, −8.799926322678350992254256043172, −8.327451658956342676376448531596, −7.12870278160581339642576039238, −6.81822840624252223887314012600, −5.25155483822064263846388472493, −4.29754274356016102503413914637, −3.42843411473319009605280687695, −2.48552900049654935566449091653, −1.42616189108533889416008998121,
1.42616189108533889416008998121, 2.48552900049654935566449091653, 3.42843411473319009605280687695, 4.29754274356016102503413914637, 5.25155483822064263846388472493, 6.81822840624252223887314012600, 7.12870278160581339642576039238, 8.327451658956342676376448531596, 8.799926322678350992254256043172, 9.359627193875159814630097624569
