L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 7-s + 8-s + 9-s + 12-s − 6·13-s + 14-s + 16-s − 6·17-s + 18-s + 2·19-s + 21-s + 24-s − 6·26-s + 27-s + 28-s − 8·29-s + 32-s − 6·34-s + 36-s − 10·37-s + 2·38-s − 6·39-s − 8·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.288·12-s − 1.66·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.458·19-s + 0.218·21-s + 0.204·24-s − 1.17·26-s + 0.192·27-s + 0.188·28-s − 1.48·29-s + 0.176·32-s − 1.02·34-s + 1/6·36-s − 1.64·37-s + 0.324·38-s − 0.960·39-s − 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 127050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.899906140\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.899906140\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−13.35572166021956, −13.10525597350238, −12.78053776537434, −11.99646756393991, −11.67348400103960, −11.30649160180620, −10.61479149834458, −10.01949018471890, −9.794484076601632, −9.023657836832391, −8.610708388699267, −8.125183950194503, −7.320576678721226, −7.121848530704416, −6.759868420351215, −5.863368464769409, −5.309827463493272, −4.861761977893239, −4.447670554031885, −3.705848901093780, −3.305306641578422, −2.476458040533114, −2.092715337559024, −1.613545438881810, −0.3970683028261127,
0.3970683028261127, 1.613545438881810, 2.092715337559024, 2.476458040533114, 3.305306641578422, 3.705848901093780, 4.447670554031885, 4.861761977893239, 5.309827463493272, 5.863368464769409, 6.759868420351215, 7.121848530704416, 7.320576678721226, 8.125183950194503, 8.610708388699267, 9.023657836832391, 9.794484076601632, 10.01949018471890, 10.61479149834458, 11.30649160180620, 11.67348400103960, 11.99646756393991, 12.78053776537434, 13.10525597350238, 13.35572166021956