| L(s) = 1 | + 2.20·2-s + 3-s + 2.87·4-s + 3.02·5-s + 2.20·6-s + 7-s + 1.93·8-s + 9-s + 6.67·10-s + 0.584·11-s + 2.87·12-s − 2.54·13-s + 2.20·14-s + 3.02·15-s − 1.47·16-s + 6.58·17-s + 2.20·18-s − 2.27·19-s + 8.69·20-s + 21-s + 1.29·22-s + 0.103·23-s + 1.93·24-s + 4.13·25-s − 5.62·26-s + 27-s + 2.87·28-s + ⋯ |
| L(s) = 1 | + 1.56·2-s + 0.577·3-s + 1.43·4-s + 1.35·5-s + 0.901·6-s + 0.377·7-s + 0.685·8-s + 0.333·9-s + 2.11·10-s + 0.176·11-s + 0.830·12-s − 0.706·13-s + 0.590·14-s + 0.780·15-s − 0.368·16-s + 1.59·17-s + 0.520·18-s − 0.521·19-s + 1.94·20-s + 0.218·21-s + 0.275·22-s + 0.0216·23-s + 0.395·24-s + 0.826·25-s − 1.10·26-s + 0.192·27-s + 0.543·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.761094868\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.761094868\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 191 | \( 1 - T \) |
| good | 2 | \( 1 - 2.20T + 2T^{2} \) |
| 5 | \( 1 - 3.02T + 5T^{2} \) |
| 11 | \( 1 - 0.584T + 11T^{2} \) |
| 13 | \( 1 + 2.54T + 13T^{2} \) |
| 17 | \( 1 - 6.58T + 17T^{2} \) |
| 19 | \( 1 + 2.27T + 19T^{2} \) |
| 23 | \( 1 - 0.103T + 23T^{2} \) |
| 29 | \( 1 - 1.55T + 29T^{2} \) |
| 31 | \( 1 - 0.308T + 31T^{2} \) |
| 37 | \( 1 - 5.81T + 37T^{2} \) |
| 41 | \( 1 + 0.802T + 41T^{2} \) |
| 43 | \( 1 + 4.76T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 - 13.9T + 53T^{2} \) |
| 59 | \( 1 + 8.65T + 59T^{2} \) |
| 61 | \( 1 - 4.76T + 61T^{2} \) |
| 67 | \( 1 - 9.29T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 - 4.17T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + 1.79T + 83T^{2} \) |
| 89 | \( 1 + 5.09T + 89T^{2} \) |
| 97 | \( 1 - 14.7T + 97T^{2} \) |
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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
−8.382606644215602604745224868073, −7.51914322793375108681005909211, −6.70002802493056293543044550710, −6.00840632693399884414953619532, −5.36300103641179696243312197591, −4.78324769978977712186646376034, −3.87929796679734572375833348958, −2.98658667066231962048751875546, −2.32869133305748730303179152979, −1.45312132695281623645885110617,
1.45312132695281623645885110617, 2.32869133305748730303179152979, 2.98658667066231962048751875546, 3.87929796679734572375833348958, 4.78324769978977712186646376034, 5.36300103641179696243312197591, 6.00840632693399884414953619532, 6.70002802493056293543044550710, 7.51914322793375108681005909211, 8.382606644215602604745224868073
