| L(s) = 1 | + 1.10·3-s − 2.70·5-s − 3.17·7-s − 1.77·9-s − 1.34·11-s − 4.41·13-s − 2.98·15-s − 0.889·17-s − 7.80·19-s − 3.51·21-s + 6.37·23-s + 2.29·25-s − 5.27·27-s + 4.35·29-s − 7.10·31-s − 1.48·33-s + 8.58·35-s + 1.46·37-s − 4.87·39-s − 9.53·41-s − 0.885·43-s + 4.80·45-s − 6.15·47-s + 3.10·49-s − 0.982·51-s − 6.04·53-s + 3.62·55-s + ⋯ |
| L(s) = 1 | + 0.637·3-s − 1.20·5-s − 1.20·7-s − 0.593·9-s − 0.404·11-s − 1.22·13-s − 0.770·15-s − 0.215·17-s − 1.79·19-s − 0.766·21-s + 1.33·23-s + 0.459·25-s − 1.01·27-s + 0.807·29-s − 1.27·31-s − 0.257·33-s + 1.45·35-s + 0.240·37-s − 0.780·39-s − 1.48·41-s − 0.135·43-s + 0.716·45-s − 0.897·47-s + 0.443·49-s − 0.137·51-s − 0.830·53-s + 0.488·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1506574147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1506574147\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 - 1.10T + 3T^{2} \) |
| 5 | \( 1 + 2.70T + 5T^{2} \) |
| 7 | \( 1 + 3.17T + 7T^{2} \) |
| 11 | \( 1 + 1.34T + 11T^{2} \) |
| 13 | \( 1 + 4.41T + 13T^{2} \) |
| 17 | \( 1 + 0.889T + 17T^{2} \) |
| 19 | \( 1 + 7.80T + 19T^{2} \) |
| 23 | \( 1 - 6.37T + 23T^{2} \) |
| 29 | \( 1 - 4.35T + 29T^{2} \) |
| 31 | \( 1 + 7.10T + 31T^{2} \) |
| 37 | \( 1 - 1.46T + 37T^{2} \) |
| 41 | \( 1 + 9.53T + 41T^{2} \) |
| 43 | \( 1 + 0.885T + 43T^{2} \) |
| 47 | \( 1 + 6.15T + 47T^{2} \) |
| 53 | \( 1 + 6.04T + 53T^{2} \) |
| 59 | \( 1 - 7.23T + 59T^{2} \) |
| 61 | \( 1 - 0.532T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 - 6.84T + 71T^{2} \) |
| 73 | \( 1 + 9.94T + 73T^{2} \) |
| 79 | \( 1 + 8.29T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 + 18.1T + 89T^{2} \) |
| 97 | \( 1 - 3.51T + 97T^{2} \) |
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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
−7.87231793345802636639179058547, −7.13621423635529543460590047071, −6.70648563236482633087606621501, −5.79203392988134579280526255769, −4.85436379389122366078339533237, −4.20555197706799009714859470490, −3.26956183848288639919095297995, −2.95623837608018386340372914403, −2.02586857249618164626534492072, −0.16961860292006294678645197529,
0.16961860292006294678645197529, 2.02586857249618164626534492072, 2.95623837608018386340372914403, 3.26956183848288639919095297995, 4.20555197706799009714859470490, 4.85436379389122366078339533237, 5.79203392988134579280526255769, 6.70648563236482633087606621501, 7.13621423635529543460590047071, 7.87231793345802636639179058547
