| L(s) = 1 | − 2.36·2-s + 0.351·3-s + 3.61·4-s − 1.64·5-s − 0.831·6-s − 1.35·7-s − 3.81·8-s − 2.87·9-s + 3.88·10-s + 11-s + 1.26·12-s − 5.03·13-s + 3.21·14-s − 0.576·15-s + 1.82·16-s − 2.16·17-s + 6.81·18-s + 1.84·19-s − 5.92·20-s − 0.476·21-s − 2.36·22-s − 1.62·23-s − 1.34·24-s − 2.30·25-s + 11.9·26-s − 2.06·27-s − 4.90·28-s + ⋯ |
| L(s) = 1 | − 1.67·2-s + 0.202·3-s + 1.80·4-s − 0.734·5-s − 0.339·6-s − 0.513·7-s − 1.34·8-s − 0.958·9-s + 1.22·10-s + 0.301·11-s + 0.366·12-s − 1.39·13-s + 0.859·14-s − 0.148·15-s + 0.455·16-s − 0.524·17-s + 1.60·18-s + 0.423·19-s − 1.32·20-s − 0.104·21-s − 0.505·22-s − 0.338·23-s − 0.273·24-s − 0.460·25-s + 2.33·26-s − 0.397·27-s − 0.927·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3370099719\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3370099719\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 2 | \( 1 + 2.36T + 2T^{2} \) |
| 3 | \( 1 - 0.351T + 3T^{2} \) |
| 5 | \( 1 + 1.64T + 5T^{2} \) |
| 7 | \( 1 + 1.35T + 7T^{2} \) |
| 13 | \( 1 + 5.03T + 13T^{2} \) |
| 17 | \( 1 + 2.16T + 17T^{2} \) |
| 19 | \( 1 - 1.84T + 19T^{2} \) |
| 23 | \( 1 + 1.62T + 23T^{2} \) |
| 29 | \( 1 - 7.80T + 29T^{2} \) |
| 31 | \( 1 + 3.31T + 31T^{2} \) |
| 37 | \( 1 + 2.36T + 37T^{2} \) |
| 41 | \( 1 - 6.18T + 41T^{2} \) |
| 43 | \( 1 - 1.24T + 43T^{2} \) |
| 47 | \( 1 + 1.52T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 - 5.80T + 59T^{2} \) |
| 61 | \( 1 - 9.41T + 61T^{2} \) |
| 67 | \( 1 - 5.76T + 67T^{2} \) |
| 71 | \( 1 + 8.95T + 71T^{2} \) |
| 73 | \( 1 + 8.92T + 73T^{2} \) |
| 79 | \( 1 + 4.64T + 79T^{2} \) |
| 83 | \( 1 + 0.995T + 83T^{2} \) |
| 89 | \( 1 + 5.76T + 89T^{2} \) |
| 97 | \( 1 - 2.61T + 97T^{2} \) |
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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
−9.488993662121094781278522994502, −8.694684742536913486640721508190, −8.133461939179648405507440368470, −7.33248965860798014562553855914, −6.75313067955267288177546895617, −5.63801684047133029204193417959, −4.32450327257727761418232489840, −3.04242187615327490284885052684, −2.18730103283553895238862100989, −0.49763966136493061064097775359,
0.49763966136493061064097775359, 2.18730103283553895238862100989, 3.04242187615327490284885052684, 4.32450327257727761418232489840, 5.63801684047133029204193417959, 6.75313067955267288177546895617, 7.33248965860798014562553855914, 8.133461939179648405507440368470, 8.694684742536913486640721508190, 9.488993662121094781278522994502
