| L(s) = 1 | − 2.53·3-s + 1.50·5-s − 0.478·7-s + 3.41·9-s − 4.20·11-s + 4.32·13-s − 3.80·15-s + 2.05·17-s − 0.535·19-s + 1.21·21-s + 6.64·23-s − 2.74·25-s − 1.05·27-s + 0.258·29-s − 4.57·31-s + 10.6·33-s − 0.719·35-s − 11.8·37-s − 10.9·39-s − 5.23·41-s + 8.69·43-s + 5.13·45-s + 11.5·47-s − 6.77·49-s − 5.20·51-s + 12.2·53-s − 6.31·55-s + ⋯ |
| L(s) = 1 | − 1.46·3-s + 0.671·5-s − 0.180·7-s + 1.13·9-s − 1.26·11-s + 1.20·13-s − 0.982·15-s + 0.498·17-s − 0.122·19-s + 0.264·21-s + 1.38·23-s − 0.548·25-s − 0.203·27-s + 0.0480·29-s − 0.822·31-s + 1.85·33-s − 0.121·35-s − 1.95·37-s − 1.75·39-s − 0.817·41-s + 1.32·43-s + 0.765·45-s + 1.68·47-s − 0.967·49-s − 0.729·51-s + 1.68·53-s − 0.851·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.052386980\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.052386980\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 + 2.53T + 3T^{2} \) |
| 5 | \( 1 - 1.50T + 5T^{2} \) |
| 7 | \( 1 + 0.478T + 7T^{2} \) |
| 11 | \( 1 + 4.20T + 11T^{2} \) |
| 13 | \( 1 - 4.32T + 13T^{2} \) |
| 17 | \( 1 - 2.05T + 17T^{2} \) |
| 19 | \( 1 + 0.535T + 19T^{2} \) |
| 23 | \( 1 - 6.64T + 23T^{2} \) |
| 29 | \( 1 - 0.258T + 29T^{2} \) |
| 31 | \( 1 + 4.57T + 31T^{2} \) |
| 37 | \( 1 + 11.8T + 37T^{2} \) |
| 41 | \( 1 + 5.23T + 41T^{2} \) |
| 43 | \( 1 - 8.69T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 + 3.96T + 59T^{2} \) |
| 61 | \( 1 - 4.23T + 61T^{2} \) |
| 67 | \( 1 + 14.8T + 67T^{2} \) |
| 71 | \( 1 - 16.3T + 71T^{2} \) |
| 73 | \( 1 - 6.11T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 - 5.39T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−8.532657252757429093043019903652, −7.48182873469392033238730406426, −6.84398005116360025701677807459, −6.02040487588617649484182052263, −5.48534156316420181771380973322, −5.11838089606976701284544314199, −3.96333410669963484838830360952, −2.94839062310871260709808245899, −1.74126316928630625337641668160, −0.64152781743006954211110076330,
0.64152781743006954211110076330, 1.74126316928630625337641668160, 2.94839062310871260709808245899, 3.96333410669963484838830360952, 5.11838089606976701284544314199, 5.48534156316420181771380973322, 6.02040487588617649484182052263, 6.84398005116360025701677807459, 7.48182873469392033238730406426, 8.532657252757429093043019903652
