| L(s) = 1 | − 0.510·2-s − 1.73·4-s − 2.52·5-s + 1.21·7-s + 1.90·8-s + 1.28·10-s + 0.502·11-s + 6.57·13-s − 0.619·14-s + 2.50·16-s + 5.32·17-s + 2.19·19-s + 4.38·20-s − 0.256·22-s + 23-s + 1.35·25-s − 3.35·26-s − 2.10·28-s + 29-s + 9.31·31-s − 5.09·32-s − 2.72·34-s − 3.05·35-s − 1.80·37-s − 1.12·38-s − 4.81·40-s − 0.987·41-s + ⋯ |
| L(s) = 1 | − 0.361·2-s − 0.869·4-s − 1.12·5-s + 0.458·7-s + 0.675·8-s + 0.407·10-s + 0.151·11-s + 1.82·13-s − 0.165·14-s + 0.625·16-s + 1.29·17-s + 0.503·19-s + 0.980·20-s − 0.0546·22-s + 0.208·23-s + 0.270·25-s − 0.658·26-s − 0.398·28-s + 0.185·29-s + 1.67·31-s − 0.901·32-s − 0.466·34-s − 0.516·35-s − 0.296·37-s − 0.182·38-s − 0.761·40-s − 0.154·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.375346377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.375346377\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 0.510T + 2T^{2} \) |
| 5 | \( 1 + 2.52T + 5T^{2} \) |
| 7 | \( 1 - 1.21T + 7T^{2} \) |
| 11 | \( 1 - 0.502T + 11T^{2} \) |
| 13 | \( 1 - 6.57T + 13T^{2} \) |
| 17 | \( 1 - 5.32T + 17T^{2} \) |
| 19 | \( 1 - 2.19T + 19T^{2} \) |
| 31 | \( 1 - 9.31T + 31T^{2} \) |
| 37 | \( 1 + 1.80T + 37T^{2} \) |
| 41 | \( 1 + 0.987T + 41T^{2} \) |
| 43 | \( 1 - 7.62T + 43T^{2} \) |
| 47 | \( 1 - 6.90T + 47T^{2} \) |
| 53 | \( 1 + 13.2T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 + 5.76T + 61T^{2} \) |
| 67 | \( 1 - 7.64T + 67T^{2} \) |
| 71 | \( 1 + 0.291T + 71T^{2} \) |
| 73 | \( 1 - 7.58T + 73T^{2} \) |
| 79 | \( 1 - 6.30T + 79T^{2} \) |
| 83 | \( 1 - 1.76T + 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 + 7.03T + 97T^{2} \) |
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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
−8.087185136322590710605244557610, −7.78613478718222763428666111074, −6.78931491643147439676751619148, −5.86947000465972711589087987711, −5.14316301751339825230478903952, −4.27604056550385289344555154612, −3.77405610343681403380225851677, −3.06539022319252783384796524155, −1.37093880512324081461755669231, −0.76939182166492468415381176489,
0.76939182166492468415381176489, 1.37093880512324081461755669231, 3.06539022319252783384796524155, 3.77405610343681403380225851677, 4.27604056550385289344555154612, 5.14316301751339825230478903952, 5.86947000465972711589087987711, 6.78931491643147439676751619148, 7.78613478718222763428666111074, 8.087185136322590710605244557610
