L(s) = 1 | − 5-s + 7-s − 2·11-s − 4·13-s + 3·17-s − 2·19-s + 6·23-s − 4·25-s + 6·29-s + 4·31-s − 35-s − 11·37-s + 5·41-s + 7·43-s + 5·47-s + 49-s − 6·53-s + 2·55-s + 9·59-s + 6·61-s + 4·65-s + 4·67-s + 8·71-s − 2·73-s − 2·77-s + 11·79-s − 5·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.377·7-s − 0.603·11-s − 1.10·13-s + 0.727·17-s − 0.458·19-s + 1.25·23-s − 4/5·25-s + 1.11·29-s + 0.718·31-s − 0.169·35-s − 1.80·37-s + 0.780·41-s + 1.06·43-s + 0.729·47-s + 1/7·49-s − 0.824·53-s + 0.269·55-s + 1.17·59-s + 0.768·61-s + 0.496·65-s + 0.488·67-s + 0.949·71-s − 0.234·73-s − 0.227·77-s + 1.23·79-s − 0.548·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.508552632\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.508552632\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−8.602613108938255410357854845912, −7.927364248707428873140249962094, −7.34076932564030184194277864610, −6.58993068392545314813136782380, −5.46485324955209523186855353004, −4.92767632072075028444970863362, −4.06108449130874599671740601455, −3.01876135593682761462208112326, −2.17903042152708749530191610187, −0.73823367139392784839361505223,
0.73823367139392784839361505223, 2.17903042152708749530191610187, 3.01876135593682761462208112326, 4.06108449130874599671740601455, 4.92767632072075028444970863362, 5.46485324955209523186855353004, 6.58993068392545314813136782380, 7.34076932564030184194277864610, 7.927364248707428873140249962094, 8.602613108938255410357854845912