| L(s) = 1 | − 27.4·3-s + 86.8·5-s + 19.8·7-s + 512.·9-s + 85.3·11-s − 229.·13-s − 2.38e3·15-s + 1.35e3·17-s + 2.79e3·19-s − 544.·21-s + 1.85e3·23-s + 4.41e3·25-s − 7.42e3·27-s + 7.31e3·29-s + 2.93e3·31-s − 2.34e3·33-s + 1.72e3·35-s + 2.57e3·37-s + 6.29e3·39-s − 3.53e3·41-s − 1.84e3·43-s + 4.45e4·45-s + 7.06e3·47-s − 1.64e4·49-s − 3.72e4·51-s − 3.85e3·53-s + 7.41e3·55-s + ⋯ |
| L(s) = 1 | − 1.76·3-s + 1.55·5-s + 0.152·7-s + 2.11·9-s + 0.212·11-s − 0.375·13-s − 2.74·15-s + 1.13·17-s + 1.77·19-s − 0.269·21-s + 0.731·23-s + 1.41·25-s − 1.95·27-s + 1.61·29-s + 0.548·31-s − 0.375·33-s + 0.237·35-s + 0.309·37-s + 0.663·39-s − 0.328·41-s − 0.152·43-s + 3.27·45-s + 0.466·47-s − 0.976·49-s − 2.00·51-s − 0.188·53-s + 0.330·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.090946164\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.090946164\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 43 | \( 1 + 1.84e3T \) |
| good | 3 | \( 1 + 27.4T + 243T^{2} \) |
| 5 | \( 1 - 86.8T + 3.12e3T^{2} \) |
| 7 | \( 1 - 19.8T + 1.68e4T^{2} \) |
| 11 | \( 1 - 85.3T + 1.61e5T^{2} \) |
| 13 | \( 1 + 229.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.35e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.79e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.85e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.31e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.93e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.57e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 3.53e3T + 1.15e8T^{2} \) |
| 47 | \( 1 - 7.06e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.85e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.79e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.92e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.48e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 8.95e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.51e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.32e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.81e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.71e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.39e4T + 8.58e9T^{2} \) |
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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
−9.965465394122618331543496709069, −9.266843972156024896354424075792, −7.69590005536953616241209919177, −6.69735202137948435822628574080, −6.04050072548953434215588414291, −5.27694519931740567705963038981, −4.78695978228313743289944498218, −3.00047529213418785977059120806, −1.46171398176113093221432422709, −0.852239871846439444138778347290,
0.852239871846439444138778347290, 1.46171398176113093221432422709, 3.00047529213418785977059120806, 4.78695978228313743289944498218, 5.27694519931740567705963038981, 6.04050072548953434215588414291, 6.69735202137948435822628574080, 7.69590005536953616241209919177, 9.266843972156024896354424075792, 9.965465394122618331543496709069
