| L(s) = 1 | − 2.56·2-s − 1.56·3-s + 4.56·4-s − 5-s + 4·6-s − 6.56·8-s − 0.561·9-s + 2.56·10-s − 1.56·11-s − 7.12·12-s − 0.438·13-s + 1.56·15-s + 7.68·16-s + 0.438·17-s + 1.43·18-s + 7.12·19-s − 4.56·20-s + 4·22-s + 3.12·23-s + 10.2·24-s + 25-s + 1.12·26-s + 5.56·27-s + 6.68·29-s − 4·30-s − 6.56·32-s + 2.43·33-s + ⋯ |
| L(s) = 1 | − 1.81·2-s − 0.901·3-s + 2.28·4-s − 0.447·5-s + 1.63·6-s − 2.31·8-s − 0.187·9-s + 0.810·10-s − 0.470·11-s − 2.05·12-s − 0.121·13-s + 0.403·15-s + 1.92·16-s + 0.106·17-s + 0.339·18-s + 1.63·19-s − 1.01·20-s + 0.852·22-s + 0.651·23-s + 2.09·24-s + 0.200·25-s + 0.220·26-s + 1.07·27-s + 1.24·29-s − 0.730·30-s − 1.15·32-s + 0.424·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3318306936\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3318306936\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 2.56T + 2T^{2} \) |
| 3 | \( 1 + 1.56T + 3T^{2} \) |
| 11 | \( 1 + 1.56T + 11T^{2} \) |
| 13 | \( 1 + 0.438T + 13T^{2} \) |
| 17 | \( 1 - 0.438T + 17T^{2} \) |
| 19 | \( 1 - 7.12T + 19T^{2} \) |
| 23 | \( 1 - 3.12T + 23T^{2} \) |
| 29 | \( 1 - 6.68T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 5.12T + 41T^{2} \) |
| 43 | \( 1 - 0.876T + 43T^{2} \) |
| 47 | \( 1 - 8.68T + 47T^{2} \) |
| 53 | \( 1 + 5.12T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 15.3T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 + 2.43T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 1.12T + 89T^{2} \) |
| 97 | \( 1 + 5.80T + 97T^{2} \) |
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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
−11.65016902251928448488266502048, −11.00999536014841663679749943218, −10.18596987098761460739555390847, −9.241269512968953197108791280327, −8.204534069241880410157454415098, −7.38602793913671059379780480666, −6.38351817577691066733008491555, −5.14643255163729445555918035432, −2.88370558660059795276118620728, −0.837178062567184828536418787995,
0.837178062567184828536418787995, 2.88370558660059795276118620728, 5.14643255163729445555918035432, 6.38351817577691066733008491555, 7.38602793913671059379780480666, 8.204534069241880410157454415098, 9.241269512968953197108791280327, 10.18596987098761460739555390847, 11.00999536014841663679749943218, 11.65016902251928448488266502048
