| L(s) = 1 | − 4·5-s − 2·11-s + 6·13-s + 4·17-s − 4·19-s + 11·25-s + 2·29-s − 8·31-s − 37-s + 2·41-s − 4·43-s + 12·47-s + 12·53-s + 8·55-s − 6·59-s + 14·61-s − 24·65-s − 12·67-s − 2·71-s + 2·73-s + 12·79-s + 12·83-s − 16·85-s − 12·89-s + 16·95-s − 10·97-s + 101-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 0.603·11-s + 1.66·13-s + 0.970·17-s − 0.917·19-s + 11/5·25-s + 0.371·29-s − 1.43·31-s − 0.164·37-s + 0.312·41-s − 0.609·43-s + 1.75·47-s + 1.64·53-s + 1.07·55-s − 0.781·59-s + 1.79·61-s − 2.97·65-s − 1.46·67-s − 0.237·71-s + 0.234·73-s + 1.35·79-s + 1.31·83-s − 1.73·85-s − 1.27·89-s + 1.64·95-s − 1.01·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.570616534\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.570616534\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.35641404879112, −12.98768324502120, −12.42776224760702, −11.99603471364188, −11.65993597211664, −10.91916045305285, −10.74437680917291, −10.41851789841234, −9.510497657605447, −8.855294168680806, −8.558454150307485, −8.038952760147355, −7.733165144085691, −7.053599092572374, −6.755300953605390, −5.841358576567243, −5.564283562487565, −4.797580242700288, −4.098514349362525, −3.847401444614386, −3.352133745541295, −2.737466314001990, −1.868955235274210, −0.9934445566280128, −0.4622533973086686,
0.4622533973086686, 0.9934445566280128, 1.868955235274210, 2.737466314001990, 3.352133745541295, 3.847401444614386, 4.098514349362525, 4.797580242700288, 5.564283562487565, 5.841358576567243, 6.755300953605390, 7.053599092572374, 7.733165144085691, 8.038952760147355, 8.558454150307485, 8.855294168680806, 9.510497657605447, 10.41851789841234, 10.74437680917291, 10.91916045305285, 11.65993597211664, 11.99603471364188, 12.42776224760702, 12.98768324502120, 13.35641404879112
