L(s) = 1 | − 3-s − 5-s + 9-s − 2·11-s + 2·13-s + 15-s − 4·19-s + 25-s − 27-s − 4·29-s − 6·31-s + 2·33-s − 8·37-s − 2·39-s − 10·41-s + 10·43-s − 45-s + 6·47-s + 10·53-s + 2·55-s + 4·57-s + 12·59-s − 14·61-s − 2·65-s + 6·67-s − 16·71-s − 2·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.603·11-s + 0.554·13-s + 0.258·15-s − 0.917·19-s + 1/5·25-s − 0.192·27-s − 0.742·29-s − 1.07·31-s + 0.348·33-s − 1.31·37-s − 0.320·39-s − 1.56·41-s + 1.52·43-s − 0.149·45-s + 0.875·47-s + 1.37·53-s + 0.269·55-s + 0.529·57-s + 1.56·59-s − 1.79·61-s − 0.248·65-s + 0.733·67-s − 1.89·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7742117181\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7742117181\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.43010894167573, −15.10287343763558, −14.42145365573261, −13.77553475086781, −13.10588859394909, −12.84900642302027, −12.06393956386995, −11.75422694488068, −10.96879988441165, −10.59930036849788, −10.24344368979881, −9.281916952219815, −8.789364292389815, −8.249753078071682, −7.478749755170294, −7.065278139359330, −6.411115124549831, −5.547112439844110, −5.389367389560440, −4.350398666178286, −3.939162731119033, −3.169985082071676, −2.245262764799713, −1.476649475614426, −0.3670319114921759,
0.3670319114921759, 1.476649475614426, 2.245262764799713, 3.169985082071676, 3.939162731119033, 4.350398666178286, 5.389367389560440, 5.547112439844110, 6.411115124549831, 7.065278139359330, 7.478749755170294, 8.249753078071682, 8.789364292389815, 9.281916952219815, 10.24344368979881, 10.59930036849788, 10.96879988441165, 11.75422694488068, 12.06393956386995, 12.84900642302027, 13.10588859394909, 13.77553475086781, 14.42145365573261, 15.10287343763558, 15.43010894167573