| L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s + 13-s + 16-s + 4·19-s − 20-s − 8·23-s + 25-s + 26-s − 6·29-s + 32-s − 10·37-s + 4·38-s − 40-s − 2·41-s − 8·46-s + 8·47-s − 7·49-s + 50-s + 52-s − 10·53-s − 6·58-s + 4·59-s + 10·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s + 0.277·13-s + 1/4·16-s + 0.917·19-s − 0.223·20-s − 1.66·23-s + 1/5·25-s + 0.196·26-s − 1.11·29-s + 0.176·32-s − 1.64·37-s + 0.648·38-s − 0.158·40-s − 0.312·41-s − 1.17·46-s + 1.16·47-s − 49-s + 0.141·50-s + 0.138·52-s − 1.37·53-s − 0.787·58-s + 0.520·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.398742785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.398742785\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 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
−12.58277755372974, −12.04789252560146, −11.80160872426616, −11.33483999755336, −10.91131038760849, −10.34020454268578, −9.927486579775904, −9.505216175263204, −8.771298815473828, −8.480433689697877, −7.831297251773766, −7.421805942620738, −7.124798299738156, −6.374647258459688, −6.022403750849718, −5.525139680175532, −4.988996095241711, −4.563813663396726, −3.859665723251378, −3.533310018909211, −3.182998169810583, −2.272104223863047, −1.898220749051187, −1.219159437164314, −0.3531118755398464,
0.3531118755398464, 1.219159437164314, 1.898220749051187, 2.272104223863047, 3.182998169810583, 3.533310018909211, 3.859665723251378, 4.563813663396726, 4.988996095241711, 5.525139680175532, 6.022403750849718, 6.374647258459688, 7.124798299738156, 7.421805942620738, 7.831297251773766, 8.480433689697877, 8.771298815473828, 9.505216175263204, 9.927486579775904, 10.34020454268578, 10.91131038760849, 11.33483999755336, 11.80160872426616, 12.04789252560146, 12.58277755372974
