| L(s) = 1 | + 1.59·3-s + 4.37·5-s − 7-s − 0.457·9-s − 11-s − 13-s + 6.96·15-s + 5.00·17-s + 5.82·19-s − 1.59·21-s + 4.91·23-s + 14.0·25-s − 5.51·27-s − 0.0873·29-s − 10.3·31-s − 1.59·33-s − 4.37·35-s − 4.36·37-s − 1.59·39-s + 3.46·41-s + 10.3·43-s − 2.00·45-s + 5.16·47-s + 49-s + 7.98·51-s + 9.50·53-s − 4.37·55-s + ⋯ |
| L(s) = 1 | + 0.920·3-s + 1.95·5-s − 0.377·7-s − 0.152·9-s − 0.301·11-s − 0.277·13-s + 1.79·15-s + 1.21·17-s + 1.33·19-s − 0.347·21-s + 1.02·23-s + 2.81·25-s − 1.06·27-s − 0.0162·29-s − 1.85·31-s − 0.277·33-s − 0.738·35-s − 0.717·37-s − 0.255·39-s + 0.541·41-s + 1.57·43-s − 0.298·45-s + 0.752·47-s + 0.142·49-s + 1.11·51-s + 1.30·53-s − 0.589·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.842527716\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.842527716\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 1.59T + 3T^{2} \) |
| 5 | \( 1 - 4.37T + 5T^{2} \) |
| 17 | \( 1 - 5.00T + 17T^{2} \) |
| 19 | \( 1 - 5.82T + 19T^{2} \) |
| 23 | \( 1 - 4.91T + 23T^{2} \) |
| 29 | \( 1 + 0.0873T + 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 + 4.36T + 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 - 5.16T + 47T^{2} \) |
| 53 | \( 1 - 9.50T + 53T^{2} \) |
| 59 | \( 1 + 12.3T + 59T^{2} \) |
| 61 | \( 1 + 7.50T + 61T^{2} \) |
| 67 | \( 1 + 5.26T + 67T^{2} \) |
| 71 | \( 1 + 15.1T + 71T^{2} \) |
| 73 | \( 1 - 2.63T + 73T^{2} \) |
| 79 | \( 1 + 4.66T + 79T^{2} \) |
| 83 | \( 1 - 18.1T + 83T^{2} \) |
| 89 | \( 1 - 0.350T + 89T^{2} \) |
| 97 | \( 1 - 6.42T + 97T^{2} \) |
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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
−8.800022357605693982244650650559, −7.52575625514002516020936511399, −7.24177588264114640313693211536, −5.88513613655159210130873099089, −5.71691899102391110861689936813, −4.90723289438651679245043983267, −3.42495265947108610722040986340, −2.89844054332735198535183219260, −2.13055483765513823235673811653, −1.15900776653224562826512213788,
1.15900776653224562826512213788, 2.13055483765513823235673811653, 2.89844054332735198535183219260, 3.42495265947108610722040986340, 4.90723289438651679245043983267, 5.71691899102391110861689936813, 5.88513613655159210130873099089, 7.24177588264114640313693211536, 7.52575625514002516020936511399, 8.800022357605693982244650650559
