| L(s) = 1 | − 2-s − 3-s + 4-s + 2.86·5-s + 6-s + 0.636·7-s − 8-s + 9-s − 2.86·10-s − 3.08·11-s − 12-s + 1.26·13-s − 0.636·14-s − 2.86·15-s + 16-s − 17-s − 18-s + 6.75·19-s + 2.86·20-s − 0.636·21-s + 3.08·22-s − 6.12·23-s + 24-s + 3.22·25-s − 1.26·26-s − 27-s + 0.636·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.28·5-s + 0.408·6-s + 0.240·7-s − 0.353·8-s + 0.333·9-s − 0.906·10-s − 0.929·11-s − 0.288·12-s + 0.350·13-s − 0.170·14-s − 0.740·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s + 1.54·19-s + 0.641·20-s − 0.138·21-s + 0.657·22-s − 1.27·23-s + 0.204·24-s + 0.644·25-s − 0.247·26-s − 0.192·27-s + 0.120·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.487636461\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.487636461\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 5 | \( 1 - 2.86T + 5T^{2} \) |
| 7 | \( 1 - 0.636T + 7T^{2} \) |
| 11 | \( 1 + 3.08T + 11T^{2} \) |
| 13 | \( 1 - 1.26T + 13T^{2} \) |
| 19 | \( 1 - 6.75T + 19T^{2} \) |
| 23 | \( 1 + 6.12T + 23T^{2} \) |
| 29 | \( 1 - 2.06T + 29T^{2} \) |
| 31 | \( 1 - 8.03T + 31T^{2} \) |
| 37 | \( 1 + 0.884T + 37T^{2} \) |
| 41 | \( 1 + 7.06T + 41T^{2} \) |
| 43 | \( 1 - 6.55T + 43T^{2} \) |
| 47 | \( 1 + 0.660T + 47T^{2} \) |
| 53 | \( 1 - 2.83T + 53T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 - 0.0139T + 67T^{2} \) |
| 71 | \( 1 - 5.73T + 71T^{2} \) |
| 73 | \( 1 - 6.30T + 73T^{2} \) |
| 79 | \( 1 - 7.12T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 - 1.32T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−7.979697953791948519279252635674, −7.53967990613489102342292132531, −6.44544914851742494345567623224, −6.12877656439867490380299178423, −5.30968991142756097713818903562, −4.78741228653744813537607398111, −3.45624326314161103962394605187, −2.47627761529744505218754265473, −1.73638030187544234009863948986, −0.74934660791775561309221091588,
0.74934660791775561309221091588, 1.73638030187544234009863948986, 2.47627761529744505218754265473, 3.45624326314161103962394605187, 4.78741228653744813537607398111, 5.30968991142756097713818903562, 6.12877656439867490380299178423, 6.44544914851742494345567623224, 7.53967990613489102342292132531, 7.979697953791948519279252635674
