| L(s) = 1 | − 1.77·2-s − 0.361·3-s + 1.16·4-s + 0.375·5-s + 0.643·6-s − 4.53·7-s + 1.48·8-s − 2.86·9-s − 0.667·10-s − 0.420·12-s − 0.269·13-s + 8.06·14-s − 0.135·15-s − 4.97·16-s + 0.0866·17-s + 5.10·18-s + 3.32·19-s + 0.436·20-s + 1.64·21-s − 3.66·23-s − 0.538·24-s − 4.85·25-s + 0.478·26-s + 2.12·27-s − 5.27·28-s − 8.24·29-s + 0.241·30-s + ⋯ |
| L(s) = 1 | − 1.25·2-s − 0.208·3-s + 0.581·4-s + 0.167·5-s + 0.262·6-s − 1.71·7-s + 0.526·8-s − 0.956·9-s − 0.210·10-s − 0.121·12-s − 0.0746·13-s + 2.15·14-s − 0.0350·15-s − 1.24·16-s + 0.0210·17-s + 1.20·18-s + 0.762·19-s + 0.0975·20-s + 0.357·21-s − 0.764·23-s − 0.109·24-s − 0.971·25-s + 0.0938·26-s + 0.408·27-s − 0.996·28-s − 1.53·29-s + 0.0440·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.04858151784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04858151784\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 1.77T + 2T^{2} \) |
| 3 | \( 1 + 0.361T + 3T^{2} \) |
| 5 | \( 1 - 0.375T + 5T^{2} \) |
| 7 | \( 1 + 4.53T + 7T^{2} \) |
| 13 | \( 1 + 0.269T + 13T^{2} \) |
| 17 | \( 1 - 0.0866T + 17T^{2} \) |
| 19 | \( 1 - 3.32T + 19T^{2} \) |
| 23 | \( 1 + 3.66T + 23T^{2} \) |
| 29 | \( 1 + 8.24T + 29T^{2} \) |
| 31 | \( 1 + 9.21T + 31T^{2} \) |
| 37 | \( 1 + 2.07T + 37T^{2} \) |
| 41 | \( 1 - 2.09T + 41T^{2} \) |
| 43 | \( 1 - 4.14T + 43T^{2} \) |
| 47 | \( 1 + 0.325T + 47T^{2} \) |
| 53 | \( 1 + 4.02T + 53T^{2} \) |
| 59 | \( 1 + 4.47T + 59T^{2} \) |
| 67 | \( 1 + 6.06T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 + 5.80T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 + 9.17T + 97T^{2} \) |
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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
−7.84258196739893734712470552582, −7.41711585092455158791581802221, −6.65424066688246652521488798366, −5.83743494724258407667424060773, −5.49196195500777782900613697046, −4.13950912255063597860852254931, −3.41906677807286645990748188816, −2.56185765859876550011405439347, −1.56412540076442606603390904371, −0.13391124353516114292107529697,
0.13391124353516114292107529697, 1.56412540076442606603390904371, 2.56185765859876550011405439347, 3.41906677807286645990748188816, 4.13950912255063597860852254931, 5.49196195500777782900613697046, 5.83743494724258407667424060773, 6.65424066688246652521488798366, 7.41711585092455158791581802221, 7.84258196739893734712470552582
