| L(s) = 1 | + 16·2-s − 184.·3-s + 256·4-s − 2.95e3·6-s + 2.40e3·7-s + 4.09e3·8-s + 1.45e4·9-s − 9.14e3·11-s − 4.73e4·12-s − 1.27e4·13-s + 3.84e4·14-s + 6.55e4·16-s + 1.95e5·17-s + 2.32e5·18-s − 1.03e4·19-s − 4.44e5·21-s − 1.46e5·22-s − 2.43e6·23-s − 7.57e5·24-s − 2.04e5·26-s + 9.54e5·27-s + 6.14e5·28-s + 4.66e6·29-s − 5.16e6·31-s + 1.04e6·32-s + 1.69e6·33-s + 3.13e6·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.31·3-s + 0.5·4-s − 0.932·6-s + 0.377·7-s + 0.353·8-s + 0.737·9-s − 0.188·11-s − 0.659·12-s − 0.123·13-s + 0.267·14-s + 0.250·16-s + 0.569·17-s + 0.521·18-s − 0.0181·19-s − 0.498·21-s − 0.133·22-s − 1.81·23-s − 0.466·24-s − 0.0876·26-s + 0.345·27-s + 0.188·28-s + 1.22·29-s − 1.00·31-s + 0.176·32-s + 0.248·33-s + 0.402·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 16T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 2.40e3T \) |
| good | 3 | \( 1 + 184.T + 1.96e4T^{2} \) |
| 11 | \( 1 + 9.14e3T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.27e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 1.95e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 1.03e4T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.43e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.66e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 5.16e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 3.21e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 6.84e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.69e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.27e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.31e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 6.40e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 9.40e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.25e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.04e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.15e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.12e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.97e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.42e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 9.71e8T + 7.60e17T^{2} \) |
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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
−9.895273377259890247953087523411, −8.357281073429693414576123586550, −7.33885395409765340591055114750, −6.28175287015971221753056641662, −5.60931910494525740033120021520, −4.80616747922663310382716077300, −3.81614590130731555526556781271, −2.38439755759142535378254811133, −1.14404807509660043955533255524, 0,
1.14404807509660043955533255524, 2.38439755759142535378254811133, 3.81614590130731555526556781271, 4.80616747922663310382716077300, 5.60931910494525740033120021520, 6.28175287015971221753056641662, 7.33885395409765340591055114750, 8.357281073429693414576123586550, 9.895273377259890247953087523411
