L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·6-s − 7-s + 8-s + 9-s − 2·12-s − 14-s + 16-s + 6·17-s + 18-s − 2·19-s + 2·21-s − 2·24-s − 5·25-s + 4·27-s − 28-s − 6·29-s + 4·31-s + 32-s + 6·34-s + 36-s − 2·37-s − 2·38-s − 6·41-s + 2·42-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.577·12-s − 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.458·19-s + 0.436·21-s − 0.408·24-s − 25-s + 0.769·27-s − 0.188·28-s − 1.11·29-s + 0.718·31-s + 0.176·32-s + 1.02·34-s + 1/6·36-s − 0.328·37-s − 0.324·38-s − 0.937·41-s + 0.308·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.648576157\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.648576157\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−9.048963436684934837350190246147, −7.985543387818251367130758863278, −7.21314366054207864456422661616, −6.39830303827760347609498636850, −5.67659105640301163348975864880, −5.32009626342602375518070026112, −4.20093704083038633652643494480, −3.42971528113344384838316462715, −2.23652265123208151083734264406, −0.78150816281722635549710851356,
0.78150816281722635549710851356, 2.23652265123208151083734264406, 3.42971528113344384838316462715, 4.20093704083038633652643494480, 5.32009626342602375518070026112, 5.67659105640301163348975864880, 6.39830303827760347609498636850, 7.21314366054207864456422661616, 7.985543387818251367130758863278, 9.048963436684934837350190246147