| L(s) = 1 | − 1.94·2-s + 3-s + 1.79·4-s + 3.69·5-s − 1.94·6-s − 2.45·7-s + 0.390·8-s + 9-s − 7.19·10-s + 4.43·11-s + 1.79·12-s − 4.28·13-s + 4.78·14-s + 3.69·15-s − 4.36·16-s − 1.49·17-s − 1.94·18-s + 1.06·19-s + 6.64·20-s − 2.45·21-s − 8.65·22-s + 23-s + 0.390·24-s + 8.63·25-s + 8.35·26-s + 27-s − 4.41·28-s + ⋯ |
| L(s) = 1 | − 1.37·2-s + 0.577·3-s + 0.899·4-s + 1.65·5-s − 0.795·6-s − 0.927·7-s + 0.138·8-s + 0.333·9-s − 2.27·10-s + 1.33·11-s + 0.519·12-s − 1.18·13-s + 1.27·14-s + 0.953·15-s − 1.09·16-s − 0.363·17-s − 0.459·18-s + 0.244·19-s + 1.48·20-s − 0.535·21-s − 1.84·22-s + 0.208·23-s + 0.0797·24-s + 1.72·25-s + 1.63·26-s + 0.192·27-s − 0.834·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.374607362\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.374607362\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 1.94T + 2T^{2} \) |
| 5 | \( 1 - 3.69T + 5T^{2} \) |
| 7 | \( 1 + 2.45T + 7T^{2} \) |
| 11 | \( 1 - 4.43T + 11T^{2} \) |
| 13 | \( 1 + 4.28T + 13T^{2} \) |
| 17 | \( 1 + 1.49T + 17T^{2} \) |
| 19 | \( 1 - 1.06T + 19T^{2} \) |
| 31 | \( 1 - 2.48T + 31T^{2} \) |
| 37 | \( 1 - 3.20T + 37T^{2} \) |
| 41 | \( 1 - 4.57T + 41T^{2} \) |
| 43 | \( 1 - 5.98T + 43T^{2} \) |
| 47 | \( 1 + 4.84T + 47T^{2} \) |
| 53 | \( 1 - 4.65T + 53T^{2} \) |
| 59 | \( 1 - 0.167T + 59T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 - 8.22T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 0.909T + 79T^{2} \) |
| 83 | \( 1 + 4.61T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 - 1.88T + 97T^{2} \) |
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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
−9.269083080400181534398535489523, −8.794035325358135392313546659950, −7.73991038964016770747260550303, −6.75350680762897632859243218425, −6.49768547426992803750253011620, −5.29696877865962356009553843056, −4.15120257172357416840094861835, −2.75461743014889725105866645127, −2.05218374751640170036086704962, −0.973548566113093204979476594524,
0.973548566113093204979476594524, 2.05218374751640170036086704962, 2.75461743014889725105866645127, 4.15120257172357416840094861835, 5.29696877865962356009553843056, 6.49768547426992803750253011620, 6.75350680762897632859243218425, 7.73991038964016770747260550303, 8.794035325358135392313546659950, 9.269083080400181534398535489523
