| L(s) = 1 | − 4·5-s − 7-s + 2·11-s + 2·13-s − 4·19-s − 6·23-s + 11·25-s − 10·29-s + 8·31-s + 4·35-s − 10·37-s + 4·41-s − 8·43-s − 4·47-s + 49-s + 10·53-s − 8·55-s − 8·59-s + 6·61-s − 8·65-s + 4·67-s + 14·71-s + 6·73-s − 2·77-s − 4·79-s + 12·83-s − 4·89-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 0.377·7-s + 0.603·11-s + 0.554·13-s − 0.917·19-s − 1.25·23-s + 11/5·25-s − 1.85·29-s + 1.43·31-s + 0.676·35-s − 1.64·37-s + 0.624·41-s − 1.21·43-s − 0.583·47-s + 1/7·49-s + 1.37·53-s − 1.07·55-s − 1.04·59-s + 0.768·61-s − 0.992·65-s + 0.488·67-s + 1.66·71-s + 0.702·73-s − 0.227·77-s − 0.450·79-s + 1.31·83-s − 0.423·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8324954468\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8324954468\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 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
−8.299884043356062889916067556533, −7.87935073079794028743168086849, −6.93317347429732963804254191787, −6.48115198191044271303057324292, −5.43540870728812758507411085485, −4.36881211524706627789595999639, −3.84265142695309311653106680052, −3.30143690281465159226599940491, −1.94242336386696047164348555559, −0.50959770511120063615442123524,
0.50959770511120063615442123524, 1.94242336386696047164348555559, 3.30143690281465159226599940491, 3.84265142695309311653106680052, 4.36881211524706627789595999639, 5.43540870728812758507411085485, 6.48115198191044271303057324292, 6.93317347429732963804254191787, 7.87935073079794028743168086849, 8.299884043356062889916067556533
