| L(s) = 1 | + 2-s − 4-s + 7-s − 3·8-s − 3·9-s + 2·11-s − 4·13-s + 14-s − 16-s + 6·17-s − 3·18-s − 8·19-s + 2·22-s − 23-s − 4·26-s − 28-s + 10·29-s + 10·31-s + 5·32-s + 6·34-s + 3·36-s − 8·37-s − 8·38-s − 2·41-s − 2·44-s − 46-s − 12·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.377·7-s − 1.06·8-s − 9-s + 0.603·11-s − 1.10·13-s + 0.267·14-s − 1/4·16-s + 1.45·17-s − 0.707·18-s − 1.83·19-s + 0.426·22-s − 0.208·23-s − 0.784·26-s − 0.188·28-s + 1.85·29-s + 1.79·31-s + 0.883·32-s + 1.02·34-s + 1/2·36-s − 1.31·37-s − 1.29·38-s − 0.312·41-s − 0.301·44-s − 0.147·46-s − 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.793527565\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.793527565\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−8.393376404030027557320054279506, −7.970958775519788007997028077879, −6.61245862411695352947689584160, −6.25752742272141619035235567335, −5.15260868143797555473704290100, −4.86586559230869122904882305727, −3.90889814446423068844497042288, −3.09514249192136623569282803930, −2.24419651153263175426309600402, −0.67536606685050410446811854895,
0.67536606685050410446811854895, 2.24419651153263175426309600402, 3.09514249192136623569282803930, 3.90889814446423068844497042288, 4.86586559230869122904882305727, 5.15260868143797555473704290100, 6.25752742272141619035235567335, 6.61245862411695352947689584160, 7.970958775519788007997028077879, 8.393376404030027557320054279506
