| L(s) = 1 | − 2-s − 2.78·3-s + 4-s + 5-s + 2.78·6-s − 3.35·7-s − 8-s + 4.73·9-s − 10-s + 3.90·11-s − 2.78·12-s − 4.76·13-s + 3.35·14-s − 2.78·15-s + 16-s − 4.89·17-s − 4.73·18-s − 2.47·19-s + 20-s + 9.32·21-s − 3.90·22-s + 6.29·23-s + 2.78·24-s + 25-s + 4.76·26-s − 4.83·27-s − 3.35·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.60·3-s + 0.5·4-s + 0.447·5-s + 1.13·6-s − 1.26·7-s − 0.353·8-s + 1.57·9-s − 0.316·10-s + 1.17·11-s − 0.802·12-s − 1.32·13-s + 0.895·14-s − 0.718·15-s + 0.250·16-s − 1.18·17-s − 1.11·18-s − 0.567·19-s + 0.223·20-s + 2.03·21-s − 0.832·22-s + 1.31·23-s + 0.567·24-s + 0.200·25-s + 0.934·26-s − 0.930·27-s − 0.633·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3313201107\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3313201107\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 + 2.78T + 3T^{2} \) |
| 7 | \( 1 + 3.35T + 7T^{2} \) |
| 11 | \( 1 - 3.90T + 11T^{2} \) |
| 13 | \( 1 + 4.76T + 13T^{2} \) |
| 17 | \( 1 + 4.89T + 17T^{2} \) |
| 19 | \( 1 + 2.47T + 19T^{2} \) |
| 23 | \( 1 - 6.29T + 23T^{2} \) |
| 29 | \( 1 + 2.05T + 29T^{2} \) |
| 31 | \( 1 + 5.06T + 31T^{2} \) |
| 37 | \( 1 - 1.59T + 37T^{2} \) |
| 41 | \( 1 + 4.70T + 41T^{2} \) |
| 43 | \( 1 + 4.61T + 43T^{2} \) |
| 47 | \( 1 + 1.83T + 47T^{2} \) |
| 53 | \( 1 - 9.36T + 53T^{2} \) |
| 59 | \( 1 - 14.0T + 59T^{2} \) |
| 61 | \( 1 + 14.8T + 61T^{2} \) |
| 67 | \( 1 + 8.21T + 67T^{2} \) |
| 71 | \( 1 + 1.00T + 71T^{2} \) |
| 73 | \( 1 + 4.30T + 73T^{2} \) |
| 79 | \( 1 - 4.93T + 79T^{2} \) |
| 83 | \( 1 - 3.19T + 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 + 0.763T + 97T^{2} \) |
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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
−8.757764683484382847843413519085, −7.32751501291498723173523976165, −6.67979722880015764118638460051, −6.57572045884688515730626780640, −5.65079511688055442715351767331, −4.90794309369470234490494917792, −3.96850100241346902470481629982, −2.76279478067677176510655357825, −1.63852898850977037978643357840, −0.39534080498663190999548091850,
0.39534080498663190999548091850, 1.63852898850977037978643357840, 2.76279478067677176510655357825, 3.96850100241346902470481629982, 4.90794309369470234490494917792, 5.65079511688055442715351767331, 6.57572045884688515730626780640, 6.67979722880015764118638460051, 7.32751501291498723173523976165, 8.757764683484382847843413519085
