| L(s) = 1 | − 2.48·2-s + 1.10·3-s + 4.16·4-s − 3.73·5-s − 2.73·6-s − 2.89·7-s − 5.37·8-s − 1.78·9-s + 9.27·10-s − 5.24·11-s + 4.59·12-s − 2.10·13-s + 7.19·14-s − 4.11·15-s + 5.01·16-s + 17-s + 4.42·18-s + 4.79·19-s − 15.5·20-s − 3.19·21-s + 13.0·22-s + 5.89·23-s − 5.93·24-s + 8.94·25-s + 5.22·26-s − 5.27·27-s − 12.0·28-s + ⋯ |
| L(s) = 1 | − 1.75·2-s + 0.636·3-s + 2.08·4-s − 1.66·5-s − 1.11·6-s − 1.09·7-s − 1.90·8-s − 0.594·9-s + 2.93·10-s − 1.58·11-s + 1.32·12-s − 0.583·13-s + 1.92·14-s − 1.06·15-s + 1.25·16-s + 0.242·17-s + 1.04·18-s + 1.09·19-s − 3.47·20-s − 0.697·21-s + 2.77·22-s + 1.22·23-s − 1.21·24-s + 1.78·25-s + 1.02·26-s − 1.01·27-s − 2.28·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2842711038\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2842711038\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 + 2.48T + 2T^{2} \) |
| 3 | \( 1 - 1.10T + 3T^{2} \) |
| 5 | \( 1 + 3.73T + 5T^{2} \) |
| 7 | \( 1 + 2.89T + 7T^{2} \) |
| 11 | \( 1 + 5.24T + 11T^{2} \) |
| 13 | \( 1 + 2.10T + 13T^{2} \) |
| 19 | \( 1 - 4.79T + 19T^{2} \) |
| 23 | \( 1 - 5.89T + 23T^{2} \) |
| 29 | \( 1 - 7.20T + 29T^{2} \) |
| 31 | \( 1 - 4.25T + 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 + 7.38T + 41T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 + 9.45T + 53T^{2} \) |
| 59 | \( 1 - 3.18T + 59T^{2} \) |
| 61 | \( 1 - 4.67T + 61T^{2} \) |
| 67 | \( 1 + 3.98T + 67T^{2} \) |
| 71 | \( 1 - 6.94T + 71T^{2} \) |
| 73 | \( 1 + 2.14T + 73T^{2} \) |
| 79 | \( 1 - 6.32T + 79T^{2} \) |
| 83 | \( 1 + 7.79T + 83T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 + 17.0T + 97T^{2} \) |
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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
−10.07750992107289667033914821512, −9.521862355747860830961800982045, −8.456115190894346837248446973757, −8.019639442103856477866288234520, −7.43629332537401373100508513994, −6.56032914272930969597557854947, −4.95313715060890635978659496668, −3.11325907145552746927606934624, −2.85713098637233739930496772789, −0.52215114753751240788379795000,
0.52215114753751240788379795000, 2.85713098637233739930496772789, 3.11325907145552746927606934624, 4.95313715060890635978659496668, 6.56032914272930969597557854947, 7.43629332537401373100508513994, 8.019639442103856477866288234520, 8.456115190894346837248446973757, 9.521862355747860830961800982045, 10.07750992107289667033914821512
