| L(s) = 1 | + 2.51·2-s + 4.32·4-s − 3.32·5-s + 2.32·7-s + 5.83·8-s − 8.34·10-s + 1.70·11-s + 4.02·13-s + 5.83·14-s + 6.02·16-s − 14.3·20-s + 4.29·22-s + 2.34·23-s + 6.02·25-s + 10.1·26-s + 10.0·28-s − 6.64·29-s + 6.70·31-s + 3.48·32-s − 7.70·35-s − 37-s − 19.3·40-s + 6.64·41-s + 0.707·43-s + 7.37·44-s + 5.90·46-s + 6·47-s + ⋯ |
| L(s) = 1 | + 1.77·2-s + 2.16·4-s − 1.48·5-s + 0.877·7-s + 2.06·8-s − 2.64·10-s + 0.514·11-s + 1.11·13-s + 1.55·14-s + 1.50·16-s − 3.20·20-s + 0.915·22-s + 0.489·23-s + 1.20·25-s + 1.98·26-s + 1.89·28-s − 1.23·29-s + 1.20·31-s + 0.616·32-s − 1.30·35-s − 0.164·37-s − 3.06·40-s + 1.03·41-s + 0.107·43-s + 1.11·44-s + 0.870·46-s + 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.330556390\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.330556390\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 2.51T + 2T^{2} \) |
| 5 | \( 1 + 3.32T + 5T^{2} \) |
| 7 | \( 1 - 2.32T + 7T^{2} \) |
| 11 | \( 1 - 1.70T + 11T^{2} \) |
| 13 | \( 1 - 4.02T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 - 2.34T + 23T^{2} \) |
| 29 | \( 1 + 6.64T + 29T^{2} \) |
| 31 | \( 1 - 6.70T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 - 6.64T + 41T^{2} \) |
| 43 | \( 1 - 0.707T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 9.96T + 53T^{2} \) |
| 59 | \( 1 + 1.70T + 59T^{2} \) |
| 61 | \( 1 - 3.38T + 61T^{2} \) |
| 67 | \( 1 + 8.37T + 67T^{2} \) |
| 71 | \( 1 - 9.41T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 + 3.34T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 - 2.67T + 89T^{2} \) |
| 97 | \( 1 - 17.7T + 97T^{2} \) |
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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
−8.337196012126021111843356662426, −7.71435956057497221456898792166, −7.01428097868081438510683021992, −6.24442470622530667956531931773, −5.39826874295739582424784674223, −4.61771212231854216260714124870, −3.94720159198705705690894644285, −3.56771318262696213402071862467, −2.44929385036094861160676087479, −1.15722349165664516667868881151,
1.15722349165664516667868881151, 2.44929385036094861160676087479, 3.56771318262696213402071862467, 3.94720159198705705690894644285, 4.61771212231854216260714124870, 5.39826874295739582424784674223, 6.24442470622530667956531931773, 7.01428097868081438510683021992, 7.71435956057497221456898792166, 8.337196012126021111843356662426
