| L(s) = 1 | − 4·2-s + 16·4-s − 61.0·5-s − 64·8-s + 244.·10-s + 36.5·11-s − 34.5·13-s + 256·16-s − 2.06e3·17-s − 452.·19-s − 977.·20-s − 146.·22-s − 1.68e3·23-s + 604.·25-s + 138.·26-s + 4.76e3·29-s + 5.26e3·31-s − 1.02e3·32-s + 8.24e3·34-s − 1.28e4·37-s + 1.80e3·38-s + 3.90e3·40-s − 7.12e3·41-s + 1.11e4·43-s + 584.·44-s + 6.73e3·46-s − 2.34e4·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.09·5-s − 0.353·8-s + 0.772·10-s + 0.0910·11-s − 0.0567·13-s + 0.250·16-s − 1.72·17-s − 0.287·19-s − 0.546·20-s − 0.0643·22-s − 0.663·23-s + 0.193·25-s + 0.0401·26-s + 1.05·29-s + 0.983·31-s − 0.176·32-s + 1.22·34-s − 1.53·37-s + 0.203·38-s + 0.386·40-s − 0.662·41-s + 0.918·43-s + 0.0455·44-s + 0.469·46-s − 1.54·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4462997418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4462997418\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 61.0T + 3.12e3T^{2} \) |
| 11 | \( 1 - 36.5T + 1.61e5T^{2} \) |
| 13 | \( 1 + 34.5T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.06e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 452.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.68e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.76e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.26e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.28e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.12e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.11e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.34e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 7.03e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.42e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.93e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.09e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.98e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.70e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.20e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.31e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.14e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.27e5T + 8.58e9T^{2} \) |
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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
−9.235278925991646510597263529544, −8.475161971037858044456120283305, −7.907157226486664259676548712457, −6.91369447500939324128543223282, −6.29119698402347659698462831588, −4.81872773617877933496246031093, −4.00686108909138917378106158666, −2.86829057405911945196047649551, −1.71387450032464521830873079177, −0.32547991289970319359220750820,
0.32547991289970319359220750820, 1.71387450032464521830873079177, 2.86829057405911945196047649551, 4.00686108909138917378106158666, 4.81872773617877933496246031093, 6.29119698402347659698462831588, 6.91369447500939324128543223282, 7.907157226486664259676548712457, 8.475161971037858044456120283305, 9.235278925991646510597263529544
