| L(s) = 1 | − 3-s + 7-s + 9-s − 4·11-s + 2·13-s + 19-s − 21-s − 5·25-s − 27-s + 8·29-s − 8·31-s + 4·33-s − 10·37-s − 2·39-s + 2·41-s + 4·43-s − 2·47-s + 49-s + 12·53-s − 57-s − 4·59-s − 10·61-s + 63-s − 8·67-s + 6·71-s + 6·73-s + 5·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.229·19-s − 0.218·21-s − 25-s − 0.192·27-s + 1.48·29-s − 1.43·31-s + 0.696·33-s − 1.64·37-s − 0.320·39-s + 0.312·41-s + 0.609·43-s − 0.291·47-s + 1/7·49-s + 1.64·53-s − 0.132·57-s − 0.520·59-s − 1.28·61-s + 0.125·63-s − 0.977·67-s + 0.712·71-s + 0.702·73-s + 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.227682736846371139757862218212, −7.54774603004303739744250845491, −6.83237077914798735999515558637, −5.82585188741248842436372335936, −5.36236997784956459297219291152, −4.52109255437283192936571124156, −3.59166116062660573192193622451, −2.50715584707627535123242612956, −1.41167507301882068370214918898, 0,
1.41167507301882068370214918898, 2.50715584707627535123242612956, 3.59166116062660573192193622451, 4.52109255437283192936571124156, 5.36236997784956459297219291152, 5.82585188741248842436372335936, 6.83237077914798735999515558637, 7.54774603004303739744250845491, 8.227682736846371139757862218212
