| L(s) = 1 | + 3-s − 1.86·5-s − 4.84·7-s + 9-s − 4.93·11-s + 2.86·13-s − 1.86·15-s + 3.10·17-s − 1.94·19-s − 4.84·21-s + 23-s − 1.51·25-s + 27-s − 29-s − 5.41·31-s − 4.93·33-s + 9.04·35-s − 6.81·37-s + 2.86·39-s + 10.1·41-s − 3.21·43-s − 1.86·45-s − 12.3·47-s + 16.4·49-s + 3.10·51-s − 13.4·53-s + 9.21·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.835·5-s − 1.83·7-s + 0.333·9-s − 1.48·11-s + 0.793·13-s − 0.482·15-s + 0.753·17-s − 0.447·19-s − 1.05·21-s + 0.208·23-s − 0.302·25-s + 0.192·27-s − 0.185·29-s − 0.973·31-s − 0.859·33-s + 1.52·35-s − 1.12·37-s + 0.458·39-s + 1.59·41-s − 0.490·43-s − 0.278·45-s − 1.79·47-s + 2.35·49-s + 0.435·51-s − 1.84·53-s + 1.24·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7438362505\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7438362505\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 + 1.86T + 5T^{2} \) |
| 7 | \( 1 + 4.84T + 7T^{2} \) |
| 11 | \( 1 + 4.93T + 11T^{2} \) |
| 13 | \( 1 - 2.86T + 13T^{2} \) |
| 17 | \( 1 - 3.10T + 17T^{2} \) |
| 19 | \( 1 + 1.94T + 19T^{2} \) |
| 31 | \( 1 + 5.41T + 31T^{2} \) |
| 37 | \( 1 + 6.81T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 + 3.21T + 43T^{2} \) |
| 47 | \( 1 + 12.3T + 47T^{2} \) |
| 53 | \( 1 + 13.4T + 53T^{2} \) |
| 59 | \( 1 - 1.37T + 59T^{2} \) |
| 61 | \( 1 + 5.44T + 61T^{2} \) |
| 67 | \( 1 + 4.14T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 + 4.90T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 + 1.89T + 89T^{2} \) |
| 97 | \( 1 - 6.06T + 97T^{2} \) |
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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
−7.74093281535400581730640030403, −7.36916111720708077418552429520, −6.44132893007584880153220621297, −5.90567194734341593221521772156, −5.00189584824769072787305900432, −4.03036013209725375481088776877, −3.21118321798125359126162283132, −3.14885013262761073997946394317, −1.90776562321353232815792303675, −0.39071070557498265506708825896,
0.39071070557498265506708825896, 1.90776562321353232815792303675, 3.14885013262761073997946394317, 3.21118321798125359126162283132, 4.03036013209725375481088776877, 5.00189584824769072787305900432, 5.90567194734341593221521772156, 6.44132893007584880153220621297, 7.36916111720708077418552429520, 7.74093281535400581730640030403
