| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s − 3·11-s − 4·13-s − 15-s + 5·19-s − 21-s − 4·23-s + 25-s + 27-s + 3·29-s − 3·33-s + 35-s − 3·37-s − 4·39-s + 7·41-s + 8·43-s − 45-s + 7·47-s − 6·49-s + 13·53-s + 3·55-s + 5·57-s − 14·59-s − 63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.904·11-s − 1.10·13-s − 0.258·15-s + 1.14·19-s − 0.218·21-s − 0.834·23-s + 1/5·25-s + 0.192·27-s + 0.557·29-s − 0.522·33-s + 0.169·35-s − 0.493·37-s − 0.640·39-s + 1.09·41-s + 1.21·43-s − 0.149·45-s + 1.02·47-s − 6/7·49-s + 1.78·53-s + 0.404·55-s + 0.662·57-s − 1.82·59-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.502082729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.502082729\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−14.13668268298842, −13.70340947005359, −13.20533891160858, −12.59225089851960, −12.14690167960189, −11.88731829588288, −11.06174934555950, −10.47899498412664, −10.15521343751821, −9.460375029242736, −9.208631417281673, −8.452276812280830, −7.876063309008428, −7.509462642603973, −7.144527206938732, −6.411365830127796, −5.592474598786057, −5.307834251937066, −4.359255523125676, −4.158620629320130, −3.165266747950722, −2.803706092577041, −2.274932452963701, −1.317933255845212, −0.3978785864571535,
0.3978785864571535, 1.317933255845212, 2.274932452963701, 2.803706092577041, 3.165266747950722, 4.158620629320130, 4.359255523125676, 5.307834251937066, 5.592474598786057, 6.411365830127796, 7.144527206938732, 7.509462642603973, 7.876063309008428, 8.452276812280830, 9.208631417281673, 9.460375029242736, 10.15521343751821, 10.47899498412664, 11.06174934555950, 11.88731829588288, 12.14690167960189, 12.59225089851960, 13.20533891160858, 13.70340947005359, 14.13668268298842
