L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 3·9-s − 4·11-s + 3·13-s + 14-s + 16-s − 17-s + 3·18-s + 4·22-s − 3·26-s − 28-s − 4·31-s − 32-s + 34-s − 3·36-s + 11·37-s − 10·41-s − 2·43-s − 4·44-s − 11·47-s + 49-s + 3·52-s + 53-s + 56-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 9-s − 1.20·11-s + 0.832·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s + 0.707·18-s + 0.852·22-s − 0.588·26-s − 0.188·28-s − 0.718·31-s − 0.176·32-s + 0.171·34-s − 1/2·36-s + 1.80·37-s − 1.56·41-s − 0.304·43-s − 0.603·44-s − 1.60·47-s + 1/7·49-s + 0.416·52-s + 0.137·53-s + 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3651020642\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3651020642\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 13 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−13.22815880795123, −12.66590012650624, −12.15979326549884, −11.47024835275461, −11.20968632573969, −10.86213435075897, −10.22359561793771, −9.858214598886502, −9.327280425453166, −8.795543400651866, −8.332330387027251, −8.016302625998569, −7.541313182967791, −6.781167572657694, −6.451927441122649, −5.838749970110652, −5.445439787661608, −4.888678561358026, −4.160205002805248, −3.338664780565563, −3.098469804766069, −2.421272133217831, −1.869021430826074, −1.054182208219933, −0.2099348975888549,
0.2099348975888549, 1.054182208219933, 1.869021430826074, 2.421272133217831, 3.098469804766069, 3.338664780565563, 4.160205002805248, 4.888678561358026, 5.445439787661608, 5.838749970110652, 6.451927441122649, 6.781167572657694, 7.541313182967791, 8.016302625998569, 8.332330387027251, 8.795543400651866, 9.327280425453166, 9.858214598886502, 10.22359561793771, 10.86213435075897, 11.20968632573969, 11.47024835275461, 12.15979326549884, 12.66590012650624, 13.22815880795123