| L(s) = 1 | + 2·2-s + 7.57·3-s + 4·4-s + 5.40·5-s + 15.1·6-s + 22.1·7-s + 8·8-s + 30.3·9-s + 10.8·10-s + 30.3·12-s − 76.8·13-s + 44.2·14-s + 40.9·15-s + 16·16-s − 59.2·17-s + 60.7·18-s − 95.2·19-s + 21.6·20-s + 167.·21-s + 142.·23-s + 60.6·24-s − 95.7·25-s − 153.·26-s + 25.6·27-s + 88.5·28-s + 20.4·29-s + 81.9·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.45·3-s + 0.5·4-s + 0.483·5-s + 1.03·6-s + 1.19·7-s + 0.353·8-s + 1.12·9-s + 0.342·10-s + 0.728·12-s − 1.63·13-s + 0.845·14-s + 0.705·15-s + 0.250·16-s − 0.845·17-s + 0.795·18-s − 1.15·19-s + 0.241·20-s + 1.74·21-s + 1.29·23-s + 0.515·24-s − 0.766·25-s − 1.15·26-s + 0.183·27-s + 0.597·28-s + 0.130·29-s + 0.498·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.841874362\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.841874362\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 7.57T + 27T^{2} \) |
| 5 | \( 1 - 5.40T + 125T^{2} \) |
| 7 | \( 1 - 22.1T + 343T^{2} \) |
| 13 | \( 1 + 76.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 59.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 95.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 142.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 20.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 213.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 145.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 82.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 151.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 90.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + 234.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 302.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 149.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 826.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 898.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 137.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 304.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 764.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 313.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 582.T + 9.12e5T^{2} \) |
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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
−11.83567776578680170671476873731, −10.71995487638473423569936727559, −9.634893086324213980677310987256, −8.625608159625335986092024845699, −7.78775151342035122360411254103, −6.76923875522082618361551891195, −5.12266871297615454111413185467, −4.25765319206792989318675068124, −2.67578560900912292113851842675, −1.94828328982834482207014141597,
1.94828328982834482207014141597, 2.67578560900912292113851842675, 4.25765319206792989318675068124, 5.12266871297615454111413185467, 6.76923875522082618361551891195, 7.78775151342035122360411254103, 8.625608159625335986092024845699, 9.634893086324213980677310987256, 10.71995487638473423569936727559, 11.83567776578680170671476873731
