| L(s) = 1 | + 2.48·2-s + 3-s + 4.17·4-s + 1.76·5-s + 2.48·6-s + 5.41·8-s + 9-s + 4.39·10-s + 0.293·11-s + 4.17·12-s + 6.33·13-s + 1.76·15-s + 5.11·16-s + 1.00·17-s + 2.48·18-s − 5.87·19-s + 7.38·20-s + 0.729·22-s + 0.838·23-s + 5.41·24-s − 1.87·25-s + 15.7·26-s + 27-s + 2.34·29-s + 4.39·30-s − 1.39·31-s + 1.86·32-s + ⋯ |
| L(s) = 1 | + 1.75·2-s + 0.577·3-s + 2.08·4-s + 0.790·5-s + 1.01·6-s + 1.91·8-s + 0.333·9-s + 1.38·10-s + 0.0885·11-s + 1.20·12-s + 1.75·13-s + 0.456·15-s + 1.27·16-s + 0.242·17-s + 0.585·18-s − 1.34·19-s + 1.65·20-s + 0.155·22-s + 0.174·23-s + 1.10·24-s − 0.375·25-s + 3.08·26-s + 0.192·27-s + 0.435·29-s + 0.802·30-s − 0.250·31-s + 0.329·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.476856111\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.476856111\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - 2.48T + 2T^{2} \) |
| 5 | \( 1 - 1.76T + 5T^{2} \) |
| 11 | \( 1 - 0.293T + 11T^{2} \) |
| 13 | \( 1 - 6.33T + 13T^{2} \) |
| 17 | \( 1 - 1.00T + 17T^{2} \) |
| 19 | \( 1 + 5.87T + 19T^{2} \) |
| 23 | \( 1 - 0.838T + 23T^{2} \) |
| 29 | \( 1 - 2.34T + 29T^{2} \) |
| 31 | \( 1 + 1.39T + 31T^{2} \) |
| 37 | \( 1 + 0.691T + 37T^{2} \) |
| 43 | \( 1 + 0.408T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 + 9.65T + 53T^{2} \) |
| 59 | \( 1 + 1.49T + 59T^{2} \) |
| 61 | \( 1 - 12.8T + 61T^{2} \) |
| 67 | \( 1 + 6.48T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 - 4.74T + 73T^{2} \) |
| 79 | \( 1 - 1.80T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 - 9.27T + 97T^{2} \) |
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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
−7.995919919054195595799924205569, −7.00473790683014119271412418118, −6.33778133102153949499737664846, −5.93922237985059309372729359578, −5.21168012910087471104546316126, −4.23843917546605044482182090818, −3.80985863377749481786727116769, −2.97705881424771570449377535926, −2.17396052842879330201523088166, −1.41081539938563807507537258002,
1.41081539938563807507537258002, 2.17396052842879330201523088166, 2.97705881424771570449377535926, 3.80985863377749481786727116769, 4.23843917546605044482182090818, 5.21168012910087471104546316126, 5.93922237985059309372729359578, 6.33778133102153949499737664846, 7.00473790683014119271412418118, 7.995919919054195595799924205569
