| L(s) = 1 | − 1.67·2-s − 3·3-s − 5.20·4-s − 6.76·5-s + 5.01·6-s − 19.0·7-s + 22.0·8-s + 9·9-s + 11.3·10-s − 65.4·11-s + 15.6·12-s − 46.8·13-s + 31.8·14-s + 20.2·15-s + 4.65·16-s + 48.0·17-s − 15.0·18-s + 147.·19-s + 35.1·20-s + 57.1·21-s + 109.·22-s + 33.1·23-s − 66.2·24-s − 79.2·25-s + 78.4·26-s − 27·27-s + 99.0·28-s + ⋯ |
| L(s) = 1 | − 0.591·2-s − 0.577·3-s − 0.650·4-s − 0.604·5-s + 0.341·6-s − 1.02·7-s + 0.976·8-s + 0.333·9-s + 0.357·10-s − 1.79·11-s + 0.375·12-s − 1.00·13-s + 0.608·14-s + 0.349·15-s + 0.0727·16-s + 0.685·17-s − 0.197·18-s + 1.77·19-s + 0.393·20-s + 0.593·21-s + 1.06·22-s + 0.300·23-s − 0.563·24-s − 0.634·25-s + 0.591·26-s − 0.192·27-s + 0.668·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3673603931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3673603931\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 59 | \( 1 + 59T \) |
| good | 2 | \( 1 + 1.67T + 8T^{2} \) |
| 5 | \( 1 + 6.76T + 125T^{2} \) |
| 7 | \( 1 + 19.0T + 343T^{2} \) |
| 11 | \( 1 + 65.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 46.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 48.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 147.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 33.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 73.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 142.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 397.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 100.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 388.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 138.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 439.T + 1.48e5T^{2} \) |
| 61 | \( 1 + 602.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 154.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 552.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 107.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 989.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 730.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.37e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 268.T + 9.12e5T^{2} \) |
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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
−12.31598702953902330189838276105, −11.09732126825105027760538073883, −9.920926305237601117865273576691, −9.608555270097395498841415858668, −7.87787655212124141879794306119, −7.46013904173703183755788536604, −5.68131609833118699741180171655, −4.69557533441361376736538183325, −3.08308300835040928772366291778, −0.51679212708775744164073075212,
0.51679212708775744164073075212, 3.08308300835040928772366291778, 4.69557533441361376736538183325, 5.68131609833118699741180171655, 7.46013904173703183755788536604, 7.87787655212124141879794306119, 9.608555270097395498841415858668, 9.920926305237601117865273576691, 11.09732126825105027760538073883, 12.31598702953902330189838276105
