L(s) = 1 | + 2-s + 4-s − 3·7-s + 8-s + 3·11-s − 13-s − 3·14-s + 16-s + 3·17-s + 3·22-s + 4·23-s − 26-s − 3·28-s − 5·29-s − 3·31-s + 32-s + 3·34-s + 12·37-s − 2·41-s − 4·43-s + 3·44-s + 4·46-s + 3·47-s + 2·49-s − 52-s + 9·53-s − 3·56-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.13·7-s + 0.353·8-s + 0.904·11-s − 0.277·13-s − 0.801·14-s + 1/4·16-s + 0.727·17-s + 0.639·22-s + 0.834·23-s − 0.196·26-s − 0.566·28-s − 0.928·29-s − 0.538·31-s + 0.176·32-s + 0.514·34-s + 1.97·37-s − 0.312·41-s − 0.609·43-s + 0.452·44-s + 0.589·46-s + 0.437·47-s + 2/7·49-s − 0.138·52-s + 1.23·53-s − 0.400·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.919982644\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.919982644\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−7.892621110288652042488521953644, −7.22681572445195529290233725233, −6.56200767706577465420344296023, −6.00449593130245924585161109271, −5.27134711941384447648237394295, −4.37504655833147681714468745295, −3.58370175708924027388339437408, −3.07148446869622941697388812954, −2.03296075777599711950431073239, −0.813520562214492182119142558246,
0.813520562214492182119142558246, 2.03296075777599711950431073239, 3.07148446869622941697388812954, 3.58370175708924027388339437408, 4.37504655833147681714468745295, 5.27134711941384447648237394295, 6.00449593130245924585161109271, 6.56200767706577465420344296023, 7.22681572445195529290233725233, 7.892621110288652042488521953644