L(s) = 1 | + 2-s + 4-s + 3·5-s + 8-s + 3·10-s − 4·13-s + 16-s + 6·17-s − 4·19-s + 3·20-s + 6·23-s + 4·25-s − 4·26-s − 3·29-s + 8·31-s + 32-s + 6·34-s + 8·37-s − 4·38-s + 3·40-s − 6·41-s + 8·43-s + 6·46-s + 6·47-s + 4·50-s − 4·52-s + 9·53-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.34·5-s + 0.353·8-s + 0.948·10-s − 1.10·13-s + 1/4·16-s + 1.45·17-s − 0.917·19-s + 0.670·20-s + 1.25·23-s + 4/5·25-s − 0.784·26-s − 0.557·29-s + 1.43·31-s + 0.176·32-s + 1.02·34-s + 1.31·37-s − 0.648·38-s + 0.474·40-s − 0.937·41-s + 1.21·43-s + 0.884·46-s + 0.875·47-s + 0.565·50-s − 0.554·52-s + 1.23·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.799773530\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.799773530\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−9.043257191829633967662307886055, −7.936219177745849931006978485567, −7.20156384395674246812312096130, −6.34847494997515706570624709902, −5.70540244651225272419030082839, −5.06588366539842132791697411033, −4.22943681392943532223456737695, −2.94884338342487937842043672252, −2.36011481689147717993766760666, −1.19829053859815870388190777931,
1.19829053859815870388190777931, 2.36011481689147717993766760666, 2.94884338342487937842043672252, 4.22943681392943532223456737695, 5.06588366539842132791697411033, 5.70540244651225272419030082839, 6.34847494997515706570624709902, 7.20156384395674246812312096130, 7.936219177745849931006978485567, 9.043257191829633967662307886055