| L(s) = 1 | + 1.92·2-s + 1.68·4-s − 2.00·5-s − 0.600·8-s − 3.85·10-s + 0.645·11-s + 0.232·13-s − 4.52·16-s + 5.71·17-s + 19-s − 3.38·20-s + 1.23·22-s − 4.21·23-s − 0.972·25-s + 0.447·26-s − 3.65·29-s − 1.75·31-s − 7.49·32-s + 10.9·34-s + 3.06·37-s + 1.92·38-s + 1.20·40-s + 6.88·41-s + 3.13·43-s + 1.08·44-s − 8.09·46-s + 12.5·47-s + ⋯ |
| L(s) = 1 | + 1.35·2-s + 0.843·4-s − 0.897·5-s − 0.212·8-s − 1.21·10-s + 0.194·11-s + 0.0646·13-s − 1.13·16-s + 1.38·17-s + 0.229·19-s − 0.757·20-s + 0.264·22-s − 0.878·23-s − 0.194·25-s + 0.0877·26-s − 0.678·29-s − 0.314·31-s − 1.32·32-s + 1.88·34-s + 0.504·37-s + 0.311·38-s + 0.190·40-s + 1.07·41-s + 0.477·43-s + 0.164·44-s − 1.19·46-s + 1.82·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.130684347\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.130684347\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 1.92T + 2T^{2} \) |
| 5 | \( 1 + 2.00T + 5T^{2} \) |
| 11 | \( 1 - 0.645T + 11T^{2} \) |
| 13 | \( 1 - 0.232T + 13T^{2} \) |
| 17 | \( 1 - 5.71T + 17T^{2} \) |
| 23 | \( 1 + 4.21T + 23T^{2} \) |
| 29 | \( 1 + 3.65T + 29T^{2} \) |
| 31 | \( 1 + 1.75T + 31T^{2} \) |
| 37 | \( 1 - 3.06T + 37T^{2} \) |
| 41 | \( 1 - 6.88T + 41T^{2} \) |
| 43 | \( 1 - 3.13T + 43T^{2} \) |
| 47 | \( 1 - 12.5T + 47T^{2} \) |
| 53 | \( 1 - 0.719T + 53T^{2} \) |
| 59 | \( 1 + 3.78T + 59T^{2} \) |
| 61 | \( 1 + 3.12T + 61T^{2} \) |
| 67 | \( 1 - 3.07T + 67T^{2} \) |
| 71 | \( 1 - 6.14T + 71T^{2} \) |
| 73 | \( 1 - 3.95T + 73T^{2} \) |
| 79 | \( 1 + 14.1T + 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 - 8.41T + 89T^{2} \) |
| 97 | \( 1 + 15.7T + 97T^{2} \) |
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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
−7.60590162035796000431790408425, −7.13463043177137143884074901941, −5.97806928997979024040308646616, −5.80901092067478088766126807824, −4.90161002595437746065167186598, −4.10926437476772647884058894933, −3.74055839257893784521660381342, −3.02731481217454499601709638176, −2.07285033900573733934324373282, −0.69243298039447954388900537831,
0.69243298039447954388900537831, 2.07285033900573733934324373282, 3.02731481217454499601709638176, 3.74055839257893784521660381342, 4.10926437476772647884058894933, 4.90161002595437746065167186598, 5.80901092067478088766126807824, 5.97806928997979024040308646616, 7.13463043177137143884074901941, 7.60590162035796000431790408425
