| L(s) = 1 | − 2-s + 4-s − 2·5-s − 8-s + 2·10-s + 11-s + 4·13-s + 16-s − 17-s + 8·19-s − 2·20-s − 22-s − 25-s − 4·26-s − 10·31-s − 32-s + 34-s + 8·37-s − 8·38-s + 2·40-s − 10·41-s − 8·43-s + 44-s + 10·47-s + 50-s + 4·52-s + 12·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.353·8-s + 0.632·10-s + 0.301·11-s + 1.10·13-s + 1/4·16-s − 0.242·17-s + 1.83·19-s − 0.447·20-s − 0.213·22-s − 1/5·25-s − 0.784·26-s − 1.79·31-s − 0.176·32-s + 0.171·34-s + 1.31·37-s − 1.29·38-s + 0.316·40-s − 1.56·41-s − 1.21·43-s + 0.150·44-s + 1.45·47-s + 0.141·50-s + 0.554·52-s + 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164934 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164934 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.403747292\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.403747292\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.27206309397659, −12.70389864162919, −12.04796254005934, −11.74155388825003, −11.33725428198980, −11.00420022436218, −10.35875552551143, −9.908841844177205, −9.315808319575579, −8.928500969771822, −8.473225554973173, −7.941390885637858, −7.418632481102285, −7.188141380056822, −6.510348514723389, −5.923597056842115, −5.439552651301259, −4.852847169350468, −4.021755551355644, −3.595087729729973, −3.278087636401757, −2.421333481035548, −1.680633433783584, −1.087554761721965, −0.4425361906634399,
0.4425361906634399, 1.087554761721965, 1.680633433783584, 2.421333481035548, 3.278087636401757, 3.595087729729973, 4.021755551355644, 4.852847169350468, 5.439552651301259, 5.923597056842115, 6.510348514723389, 7.188141380056822, 7.418632481102285, 7.941390885637858, 8.473225554973173, 8.928500969771822, 9.315808319575579, 9.908841844177205, 10.35875552551143, 11.00420022436218, 11.33725428198980, 11.74155388825003, 12.04796254005934, 12.70389864162919, 13.27206309397659
