L(s) = 1 | + 2-s + 4-s − 2·5-s + 7-s + 8-s − 3·9-s − 2·10-s + 4·11-s + 4·13-s + 14-s + 16-s + 8·17-s − 3·18-s + 2·19-s − 2·20-s + 4·22-s + 23-s − 25-s + 4·26-s + 28-s + 6·31-s + 32-s + 8·34-s − 2·35-s − 3·36-s + 10·37-s + 2·38-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.377·7-s + 0.353·8-s − 9-s − 0.632·10-s + 1.20·11-s + 1.10·13-s + 0.267·14-s + 1/4·16-s + 1.94·17-s − 0.707·18-s + 0.458·19-s − 0.447·20-s + 0.852·22-s + 0.208·23-s − 1/5·25-s + 0.784·26-s + 0.188·28-s + 1.07·31-s + 0.176·32-s + 1.37·34-s − 0.338·35-s − 1/2·36-s + 1.64·37-s + 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270802 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270802 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.399807840\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.399807840\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−12.53619926642251, −12.29715362929062, −11.80095415558396, −11.51902328182734, −11.18356935182478, −10.75589965370694, −9.935474786626414, −9.674448111247521, −8.954092778644945, −8.458371540419980, −8.061039344651493, −7.697960478759276, −7.156937715346462, −6.520302502072632, −5.963734238255269, −5.763273344880156, −5.159632125741088, −4.408905852715877, −4.119783309730074, −3.511060536397332, −3.129302310650488, −2.702169391566952, −1.636109234253045, −1.181998722164479, −0.6144922462138854,
0.6144922462138854, 1.181998722164479, 1.636109234253045, 2.702169391566952, 3.129302310650488, 3.511060536397332, 4.119783309730074, 4.408905852715877, 5.159632125741088, 5.763273344880156, 5.963734238255269, 6.520302502072632, 7.156937715346462, 7.697960478759276, 8.061039344651493, 8.458371540419980, 8.954092778644945, 9.674448111247521, 9.935474786626414, 10.75589965370694, 11.18356935182478, 11.51902328182734, 11.80095415558396, 12.29715362929062, 12.53619926642251