| L(s) = 1 | + 3.78·2-s + 5.19·3-s + 6.33·4-s + 4.98·5-s + 19.6·6-s + 7·7-s − 6.30·8-s − 0.0513·9-s + 18.8·10-s + 53.2·11-s + 32.8·12-s − 64.5·13-s + 26.5·14-s + 25.8·15-s − 74.5·16-s + 17·17-s − 0.194·18-s − 35.0·19-s + 31.5·20-s + 36.3·21-s + 201.·22-s + 146.·23-s − 32.7·24-s − 100.·25-s − 244.·26-s − 140.·27-s + 44.3·28-s + ⋯ |
| L(s) = 1 | + 1.33·2-s + 0.999·3-s + 0.791·4-s + 0.445·5-s + 1.33·6-s + 0.377·7-s − 0.278·8-s − 0.00190·9-s + 0.596·10-s + 1.46·11-s + 0.791·12-s − 1.37·13-s + 0.505·14-s + 0.445·15-s − 1.16·16-s + 0.242·17-s − 0.00254·18-s − 0.423·19-s + 0.353·20-s + 0.377·21-s + 1.95·22-s + 1.32·23-s − 0.278·24-s − 0.801·25-s − 1.84·26-s − 1.00·27-s + 0.299·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.154587298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.154587298\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 7T \) |
| 17 | \( 1 - 17T \) |
| good | 2 | \( 1 - 3.78T + 8T^{2} \) |
| 3 | \( 1 - 5.19T + 27T^{2} \) |
| 5 | \( 1 - 4.98T + 125T^{2} \) |
| 11 | \( 1 - 53.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 64.5T + 2.19e3T^{2} \) |
| 19 | \( 1 + 35.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 146.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 278.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 282.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 257.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 475.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 41.6T + 7.95e4T^{2} \) |
| 47 | \( 1 + 173.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 57.2T + 1.48e5T^{2} \) |
| 59 | \( 1 - 571.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 467.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 765.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 71.6T + 3.57e5T^{2} \) |
| 73 | \( 1 - 159.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.24e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 90.8T + 5.71e5T^{2} \) |
| 89 | \( 1 - 88.6T + 7.04e5T^{2} \) |
| 97 | \( 1 - 627.T + 9.12e5T^{2} \) |
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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
−13.30577092808613116822884850194, −12.24879380499814796356676768826, −11.37600182546143637057867408996, −9.604435227687159462187112795661, −8.875260934676640177976858922084, −7.35953008147886391413347852321, −6.03214628982971888594677580822, −4.74056086565849536828522524687, −3.51951577532816389959312990694, −2.22300531362943512228173521293,
2.22300531362943512228173521293, 3.51951577532816389959312990694, 4.74056086565849536828522524687, 6.03214628982971888594677580822, 7.35953008147886391413347852321, 8.875260934676640177976858922084, 9.604435227687159462187112795661, 11.37600182546143637057867408996, 12.24879380499814796356676768826, 13.30577092808613116822884850194
