| L(s) = 1 | − 2.24·3-s − 1.35·7-s + 2.04·9-s − 4.85·11-s + 0.198·13-s − 1.13·17-s + 19-s + 3.04·21-s − 2.55·23-s + 2.13·27-s − 10.2·29-s − 2.51·31-s + 10.8·33-s − 0.137·37-s − 0.445·39-s − 11.7·41-s + 7.59·43-s − 2.69·47-s − 5.15·49-s + 2.55·51-s + 12.8·53-s − 2.24·57-s − 5.82·59-s − 7.58·61-s − 2.78·63-s − 8.01·67-s + 5.74·69-s + ⋯ |
| L(s) = 1 | − 1.29·3-s − 0.512·7-s + 0.682·9-s − 1.46·11-s + 0.0549·13-s − 0.275·17-s + 0.229·19-s + 0.665·21-s − 0.532·23-s + 0.411·27-s − 1.90·29-s − 0.451·31-s + 1.89·33-s − 0.0225·37-s − 0.0712·39-s − 1.83·41-s + 1.15·43-s − 0.392·47-s − 0.736·49-s + 0.357·51-s + 1.76·53-s − 0.297·57-s − 0.758·59-s − 0.970·61-s − 0.350·63-s − 0.979·67-s + 0.691·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2375157693\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2375157693\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2.24T + 3T^{2} \) |
| 7 | \( 1 + 1.35T + 7T^{2} \) |
| 11 | \( 1 + 4.85T + 11T^{2} \) |
| 13 | \( 1 - 0.198T + 13T^{2} \) |
| 17 | \( 1 + 1.13T + 17T^{2} \) |
| 23 | \( 1 + 2.55T + 23T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 + 2.51T + 31T^{2} \) |
| 37 | \( 1 + 0.137T + 37T^{2} \) |
| 41 | \( 1 + 11.7T + 41T^{2} \) |
| 43 | \( 1 - 7.59T + 43T^{2} \) |
| 47 | \( 1 + 2.69T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 + 5.82T + 59T^{2} \) |
| 61 | \( 1 + 7.58T + 61T^{2} \) |
| 67 | \( 1 + 8.01T + 67T^{2} \) |
| 71 | \( 1 - 8.82T + 71T^{2} \) |
| 73 | \( 1 + 11.9T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 + 3.77T + 83T^{2} \) |
| 89 | \( 1 - 9.36T + 89T^{2} \) |
| 97 | \( 1 - 0.198T + 97T^{2} \) |
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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
−7.65345375896336755331910080119, −7.16939373874245392162709350289, −6.32531326096227138529438222620, −5.69453255930657845758254660014, −5.29984242434096859694259337498, −4.52611550644990485176532670137, −3.57624151054453288068832248097, −2.69921883977485937099238038507, −1.66511056360140578273954791080, −0.25286689369454940089851824388,
0.25286689369454940089851824388, 1.66511056360140578273954791080, 2.69921883977485937099238038507, 3.57624151054453288068832248097, 4.52611550644990485176532670137, 5.29984242434096859694259337498, 5.69453255930657845758254660014, 6.32531326096227138529438222620, 7.16939373874245392162709350289, 7.65345375896336755331910080119
