| L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s − 4·11-s + 2·13-s + 16-s − 6·17-s − 4·19-s + 20-s + 4·22-s + 23-s + 25-s − 2·26-s + 2·29-s − 32-s + 6·34-s − 2·37-s + 4·38-s − 40-s + 10·41-s − 4·43-s − 4·44-s − 46-s − 50-s + 2·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s − 1.20·11-s + 0.554·13-s + 1/4·16-s − 1.45·17-s − 0.917·19-s + 0.223·20-s + 0.852·22-s + 0.208·23-s + 1/5·25-s − 0.392·26-s + 0.371·29-s − 0.176·32-s + 1.02·34-s − 0.328·37-s + 0.648·38-s − 0.158·40-s + 1.56·41-s − 0.609·43-s − 0.603·44-s − 0.147·46-s − 0.141·50-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 101430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.93057064782458, −13.37859355619611, −12.91468267376231, −12.77905836275113, −11.89866726739295, −11.39636001629596, −10.84456340661135, −10.56534127875724, −10.17915931462649, −9.416332450701335, −9.063210722319542, −8.507569211451803, −8.154358451278309, −7.534153066916943, −6.943134390118439, −6.463506738626397, −5.954989053833726, −5.448402166120509, −4.635819178428376, −4.329695287419866, −3.355414779245549, −2.766039438793998, −2.187874740955043, −1.716953921282689, −0.7431787235417628, 0,
0.7431787235417628, 1.716953921282689, 2.187874740955043, 2.766039438793998, 3.355414779245549, 4.329695287419866, 4.635819178428376, 5.448402166120509, 5.954989053833726, 6.463506738626397, 6.943134390118439, 7.534153066916943, 8.154358451278309, 8.507569211451803, 9.063210722319542, 9.416332450701335, 10.17915931462649, 10.56534127875724, 10.84456340661135, 11.39636001629596, 11.89866726739295, 12.77905836275113, 12.91468267376231, 13.37859355619611, 13.93057064782458
