| L(s) = 1 | − 1.94·2-s + 3-s + 1.79·4-s − 3.71·5-s − 1.94·6-s − 7-s + 0.403·8-s + 9-s + 7.22·10-s + 5.06·11-s + 1.79·12-s + 3.43·13-s + 1.94·14-s − 3.71·15-s − 4.37·16-s − 2.10·17-s − 1.94·18-s − 6.45·19-s − 6.65·20-s − 21-s − 9.86·22-s + 2.14·23-s + 0.403·24-s + 8.76·25-s − 6.68·26-s + 27-s − 1.79·28-s + ⋯ |
| L(s) = 1 | − 1.37·2-s + 0.577·3-s + 0.896·4-s − 1.65·5-s − 0.795·6-s − 0.377·7-s + 0.142·8-s + 0.333·9-s + 2.28·10-s + 1.52·11-s + 0.517·12-s + 0.952·13-s + 0.520·14-s − 0.958·15-s − 1.09·16-s − 0.511·17-s − 0.459·18-s − 1.48·19-s − 1.48·20-s − 0.218·21-s − 2.10·22-s + 0.446·23-s + 0.0822·24-s + 1.75·25-s − 1.31·26-s + 0.192·27-s − 0.338·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7808703729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7808703729\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 + 1.94T + 2T^{2} \) |
| 5 | \( 1 + 3.71T + 5T^{2} \) |
| 11 | \( 1 - 5.06T + 11T^{2} \) |
| 13 | \( 1 - 3.43T + 13T^{2} \) |
| 17 | \( 1 + 2.10T + 17T^{2} \) |
| 19 | \( 1 + 6.45T + 19T^{2} \) |
| 23 | \( 1 - 2.14T + 23T^{2} \) |
| 29 | \( 1 + 1.41T + 29T^{2} \) |
| 31 | \( 1 - 6.38T + 31T^{2} \) |
| 37 | \( 1 - 0.0418T + 37T^{2} \) |
| 41 | \( 1 - 7.08T + 41T^{2} \) |
| 43 | \( 1 + 5.18T + 43T^{2} \) |
| 47 | \( 1 - 2.63T + 47T^{2} \) |
| 53 | \( 1 - 5.53T + 53T^{2} \) |
| 59 | \( 1 + 4.96T + 59T^{2} \) |
| 61 | \( 1 - 7.20T + 61T^{2} \) |
| 67 | \( 1 + 7.34T + 67T^{2} \) |
| 71 | \( 1 + 3.44T + 71T^{2} \) |
| 73 | \( 1 + 0.621T + 73T^{2} \) |
| 79 | \( 1 - 4.87T + 79T^{2} \) |
| 83 | \( 1 - 7.84T + 83T^{2} \) |
| 89 | \( 1 - 5.47T + 89T^{2} \) |
| 97 | \( 1 - 7.58T + 97T^{2} \) |
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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
−8.007345871956984984351462503499, −7.41340143815709976594071870401, −6.66375394008972165516984602277, −6.33065706729286666429039847713, −4.64549820698203757197788240432, −4.07175811002400817084899037387, −3.63167828662760306504377101310, −2.53071378212575570689842482021, −1.40857632511211534272347482591, −0.57370913430493907302621234551,
0.57370913430493907302621234551, 1.40857632511211534272347482591, 2.53071378212575570689842482021, 3.63167828662760306504377101310, 4.07175811002400817084899037387, 4.64549820698203757197788240432, 6.33065706729286666429039847713, 6.66375394008972165516984602277, 7.41340143815709976594071870401, 8.007345871956984984351462503499
