| L(s) = 1 | − 3-s − 4·5-s + 9-s + 2·11-s + 4·15-s + 17-s + 2·19-s + 11·25-s − 27-s − 2·29-s + 8·31-s − 2·33-s + 10·37-s + 12·41-s − 4·43-s − 4·45-s − 12·47-s − 7·49-s − 51-s − 2·53-s − 8·55-s − 2·57-s + 6·61-s + 14·67-s + 10·71-s − 10·73-s − 11·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.78·5-s + 1/3·9-s + 0.603·11-s + 1.03·15-s + 0.242·17-s + 0.458·19-s + 11/5·25-s − 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.348·33-s + 1.64·37-s + 1.87·41-s − 0.609·43-s − 0.596·45-s − 1.75·47-s − 49-s − 0.140·51-s − 0.274·53-s − 1.07·55-s − 0.264·57-s + 0.768·61-s + 1.71·67-s + 1.18·71-s − 1.17·73-s − 1.27·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 137904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.657418706\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.657418706\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−13.17895651395109, −12.82158505075253, −12.43333558503807, −11.76214197784631, −11.56930429589709, −11.21679455691183, −10.81635646868063, −9.940955353421601, −9.703633613386857, −9.048859631273913, −8.370035083012706, −7.895170071105827, −7.741042352622251, −6.969136373282267, −6.568446944769187, −6.101088514698886, −5.303793137636969, −4.775252148623994, −4.282988368922575, −3.895867149505848, −3.235309996384546, −2.768250125856646, −1.749973535423950, −0.8511766648638657, −0.5633645683925628,
0.5633645683925628, 0.8511766648638657, 1.749973535423950, 2.768250125856646, 3.235309996384546, 3.895867149505848, 4.282988368922575, 4.775252148623994, 5.303793137636969, 6.101088514698886, 6.568446944769187, 6.969136373282267, 7.741042352622251, 7.895170071105827, 8.370035083012706, 9.048859631273913, 9.703633613386857, 9.940955353421601, 10.81635646868063, 11.21679455691183, 11.56930429589709, 11.76214197784631, 12.43333558503807, 12.82158505075253, 13.17895651395109
