| L(s) = 1 | − 9·3-s + 59.1·5-s + 213.·7-s + 81·9-s − 126.·11-s − 884.·13-s − 532.·15-s − 1.17e3·17-s + 1.86e3·19-s − 1.91e3·21-s − 529·23-s + 369.·25-s − 729·27-s − 6.78e3·29-s + 5.14e3·31-s + 1.13e3·33-s + 1.26e4·35-s + 5.13e3·37-s + 7.96e3·39-s + 1.24e4·41-s − 4.19e3·43-s + 4.78e3·45-s − 2.30e4·47-s + 2.87e4·49-s + 1.06e4·51-s + 2.51e4·53-s − 7.47e3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.05·5-s + 1.64·7-s + 0.333·9-s − 0.315·11-s − 1.45·13-s − 0.610·15-s − 0.990·17-s + 1.18·19-s − 0.950·21-s − 0.208·23-s + 0.118·25-s − 0.192·27-s − 1.49·29-s + 0.961·31-s + 0.182·33-s + 1.74·35-s + 0.616·37-s + 0.838·39-s + 1.15·41-s − 0.346·43-s + 0.352·45-s − 1.51·47-s + 1.70·49-s + 0.571·51-s + 1.23·53-s − 0.333·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.619745701\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.619745701\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 9T \) |
| 23 | \( 1 + 529T \) |
| good | 5 | \( 1 - 59.1T + 3.12e3T^{2} \) |
| 7 | \( 1 - 213.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 126.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 884.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.17e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.86e3T + 2.47e6T^{2} \) |
| 29 | \( 1 + 6.78e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.14e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.13e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.24e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.19e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.30e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.51e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.71e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.64e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.43e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.56e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.39e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.66e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.66e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.00e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.46e4T + 8.58e9T^{2} \) |
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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
−9.376837489818463934051585563115, −8.155780508294512096787089533567, −7.50279976997768361363030323145, −6.58303243157568296666509564246, −5.34728356545657878823304812281, −5.20093306216359464471033720484, −4.17947168033845633895659373356, −2.43246183019761534113611735159, −1.86775682483801940134749855614, −0.71626281863345919185195161957,
0.71626281863345919185195161957, 1.86775682483801940134749855614, 2.43246183019761534113611735159, 4.17947168033845633895659373356, 5.20093306216359464471033720484, 5.34728356545657878823304812281, 6.58303243157568296666509564246, 7.50279976997768361363030323145, 8.155780508294512096787089533567, 9.376837489818463934051585563115
