| L(s) = 1 | − 3-s + 3·7-s − 4·9-s − 2·11-s + 2·13-s + 17-s + 12·19-s − 3·21-s − 3·23-s + 6·27-s + 29-s + 4·31-s + 2·33-s − 4·37-s − 2·39-s − 8·41-s + 4·43-s − 6·49-s − 51-s + 5·53-s − 12·57-s + 8·59-s − 15·61-s − 12·63-s + 24·67-s + 3·69-s + 6·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.13·7-s − 4/3·9-s − 0.603·11-s + 0.554·13-s + 0.242·17-s + 2.75·19-s − 0.654·21-s − 0.625·23-s + 1.15·27-s + 0.185·29-s + 0.718·31-s + 0.348·33-s − 0.657·37-s − 0.320·39-s − 1.24·41-s + 0.609·43-s − 6/7·49-s − 0.140·51-s + 0.686·53-s − 1.58·57-s + 1.04·59-s − 1.92·61-s − 1.51·63-s + 2.93·67-s + 0.361·69-s + 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.801080465\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.801080465\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 3 T + 15 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 12 T + 69 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 47 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - T - 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 73 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 53 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 89 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 5 T + 111 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 89 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 15 T + 147 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_4$ | \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 21 T + 255 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 17 T + 229 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 7 T + 77 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 13 T + 159 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 23 T + 325 T^{2} - 23 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.549086188306114807515407536749, −7.991832660371172696078132713467, −7.897469879952028997574627456276, −7.73632639063527919217083067816, −7.01461823615116209511424022975, −6.68809024919949034091786092554, −6.35840694464300036870193010589, −5.86719868972475067256547844728, −5.36012444384650687012821119510, −5.29962976239603760221876494579, −5.07042465977381800089299119960, −4.67132293343261082011012416090, −3.85784260033273398136930676320, −3.63079319877428014649182622740, −2.97980383295172907167557654382, −2.92353733152213655633838889683, −2.03583104278383959820010443173, −1.77397178223338331257220655255, −0.802396854074297544795514332761, −0.69378144217976405660290606947,
0.69378144217976405660290606947, 0.802396854074297544795514332761, 1.77397178223338331257220655255, 2.03583104278383959820010443173, 2.92353733152213655633838889683, 2.97980383295172907167557654382, 3.63079319877428014649182622740, 3.85784260033273398136930676320, 4.67132293343261082011012416090, 5.07042465977381800089299119960, 5.29962976239603760221876494579, 5.36012444384650687012821119510, 5.86719868972475067256547844728, 6.35840694464300036870193010589, 6.68809024919949034091786092554, 7.01461823615116209511424022975, 7.73632639063527919217083067816, 7.897469879952028997574627456276, 7.991832660371172696078132713467, 8.549086188306114807515407536749
