L(s) = 1 | + 1.61·2-s + 1.61·4-s − 7-s + 8-s + 9-s − 1.61·14-s + 1.61·18-s − 1.61·23-s + 25-s − 1.61·28-s − 0.618·29-s − 32-s + 1.61·36-s − 1.61·37-s − 0.618·43-s − 2.61·46-s + 49-s + 1.61·50-s + 0.618·53-s − 56-s − 1.00·58-s − 63-s − 1.61·64-s + 0.618·67-s + 0.618·71-s + 72-s − 2.61·74-s + ⋯ |
L(s) = 1 | + 1.61·2-s + 1.61·4-s − 7-s + 8-s + 9-s − 1.61·14-s + 1.61·18-s − 1.61·23-s + 25-s − 1.61·28-s − 0.618·29-s − 32-s + 1.61·36-s − 1.61·37-s − 0.618·43-s − 2.61·46-s + 49-s + 1.61·50-s + 0.618·53-s − 56-s − 1.00·58-s − 63-s − 1.61·64-s + 0.618·67-s + 0.618·71-s + 72-s − 2.61·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.140652508\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.140652508\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 1.61T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 1.61T + T^{2} \) |
| 29 | \( 1 + 0.618T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 1.61T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 0.618T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 0.618T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 0.618T + T^{2} \) |
| 71 | \( 1 - 0.618T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 0.618T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.45851621024909772695187898264, −9.816499517200323577202776028840, −8.728839033689864251438524661927, −7.36191382478388408458457980252, −6.70448672890854957653081139844, −5.93654984027054944639437756276, −4.97746569339534258684655408429, −4.00320039925494911070552056190, −3.33936229647190113630910749550, −2.06479344706385911278383389083,
2.06479344706385911278383389083, 3.33936229647190113630910749550, 4.00320039925494911070552056190, 4.97746569339534258684655408429, 5.93654984027054944639437756276, 6.70448672890854957653081139844, 7.36191382478388408458457980252, 8.728839033689864251438524661927, 9.816499517200323577202776028840, 10.45851621024909772695187898264