| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 3·11-s − 13-s + 14-s + 16-s + 3·17-s − 7·19-s + 3·22-s − 3·23-s − 5·25-s + 26-s − 28-s + 2·31-s − 32-s − 3·34-s + 37-s + 7·38-s + 6·41-s − 4·43-s − 3·44-s + 3·46-s − 6·47-s − 6·49-s + 5·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 0.904·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s − 1.60·19-s + 0.639·22-s − 0.625·23-s − 25-s + 0.196·26-s − 0.188·28-s + 0.359·31-s − 0.176·32-s − 0.514·34-s + 0.164·37-s + 1.13·38-s + 0.937·41-s − 0.609·43-s − 0.452·44-s + 0.442·46-s − 0.875·47-s − 6/7·49-s + 0.707·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 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
−10.06815972051721250592457118805, −9.339371053419225859875408435925, −8.208412645914458611430202370847, −7.73927537740876447103082520863, −6.55620004019545166157451145997, −5.78215828046120660270620191902, −4.49624163661933276807437922670, −3.14043362082132550386338477310, −1.96278840753819796229532576491, 0,
1.96278840753819796229532576491, 3.14043362082132550386338477310, 4.49624163661933276807437922670, 5.78215828046120660270620191902, 6.55620004019545166157451145997, 7.73927537740876447103082520863, 8.208412645914458611430202370847, 9.339371053419225859875408435925, 10.06815972051721250592457118805
