| L(s) = 1 | − 1.39·2-s − 0.0513·4-s + 3.05·5-s + 2.86·8-s − 4.25·10-s − 5.31·11-s + 3.31·13-s − 3.89·16-s + 0.948·17-s + 19-s − 0.156·20-s + 7.41·22-s − 1.31·23-s + 4.31·25-s − 4.62·26-s + 7.84·29-s + 4.79·31-s − 0.290·32-s − 1.32·34-s − 8.62·37-s − 1.39·38-s + 8.73·40-s + 11.3·41-s + 3.20·43-s + 0.272·44-s + 1.82·46-s − 5.84·47-s + ⋯ |
| L(s) = 1 | − 0.987·2-s − 0.0256·4-s + 1.36·5-s + 1.01·8-s − 1.34·10-s − 1.60·11-s + 0.918·13-s − 0.973·16-s + 0.230·17-s + 0.229·19-s − 0.0350·20-s + 1.58·22-s − 0.273·23-s + 0.862·25-s − 0.906·26-s + 1.45·29-s + 0.860·31-s − 0.0513·32-s − 0.227·34-s − 1.41·37-s − 0.226·38-s + 1.38·40-s + 1.76·41-s + 0.489·43-s + 0.0411·44-s + 0.269·46-s − 0.852·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.351064052\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.351064052\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 1.39T + 2T^{2} \) |
| 5 | \( 1 - 3.05T + 5T^{2} \) |
| 11 | \( 1 + 5.31T + 11T^{2} \) |
| 13 | \( 1 - 3.31T + 13T^{2} \) |
| 17 | \( 1 - 0.948T + 17T^{2} \) |
| 23 | \( 1 + 1.31T + 23T^{2} \) |
| 29 | \( 1 - 7.84T + 29T^{2} \) |
| 31 | \( 1 - 4.79T + 31T^{2} \) |
| 37 | \( 1 + 8.62T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 - 3.20T + 43T^{2} \) |
| 47 | \( 1 + 5.84T + 47T^{2} \) |
| 53 | \( 1 - 2.77T + 53T^{2} \) |
| 59 | \( 1 + 8.20T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 + 4.51T + 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 - 4.10T + 73T^{2} \) |
| 79 | \( 1 + 8.20T + 79T^{2} \) |
| 83 | \( 1 + 6.36T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−7.959419123206823320520030643860, −7.34147212169048069406056249531, −6.36189334332034868921963090791, −5.83831624011256005615186584212, −5.06867335727920224566554716107, −4.49043558764623789870469265096, −3.22425816374626620092904113684, −2.41103761521627434800362588944, −1.60207718185502690563909520429, −0.69244756312579631093206668898,
0.69244756312579631093206668898, 1.60207718185502690563909520429, 2.41103761521627434800362588944, 3.22425816374626620092904113684, 4.49043558764623789870469265096, 5.06867335727920224566554716107, 5.83831624011256005615186584212, 6.36189334332034868921963090791, 7.34147212169048069406056249531, 7.959419123206823320520030643860
