| L(s) = 1 | + 1.98·2-s − 3-s + 1.93·4-s − 1.98·6-s − 2.16·7-s − 0.119·8-s + 9-s + 3.44·11-s − 1.93·12-s + 3.74·13-s − 4.29·14-s − 4.11·16-s − 4.33·17-s + 1.98·18-s + 3.90·19-s + 2.16·21-s + 6.84·22-s + 3.78·23-s + 0.119·24-s + 7.42·26-s − 27-s − 4.20·28-s + 29-s + 10.3·31-s − 7.93·32-s − 3.44·33-s − 8.60·34-s + ⋯ |
| L(s) = 1 | + 1.40·2-s − 0.577·3-s + 0.969·4-s − 0.810·6-s − 0.818·7-s − 0.0423·8-s + 0.333·9-s + 1.03·11-s − 0.559·12-s + 1.03·13-s − 1.14·14-s − 1.02·16-s − 1.05·17-s + 0.467·18-s + 0.895·19-s + 0.472·21-s + 1.45·22-s + 0.789·23-s + 0.0244·24-s + 1.45·26-s − 0.192·27-s − 0.794·28-s + 0.185·29-s + 1.85·31-s − 1.40·32-s − 0.600·33-s − 1.47·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.003836522\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.003836522\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 1.98T + 2T^{2} \) |
| 7 | \( 1 + 2.16T + 7T^{2} \) |
| 11 | \( 1 - 3.44T + 11T^{2} \) |
| 13 | \( 1 - 3.74T + 13T^{2} \) |
| 17 | \( 1 + 4.33T + 17T^{2} \) |
| 19 | \( 1 - 3.90T + 19T^{2} \) |
| 23 | \( 1 - 3.78T + 23T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 - 7.70T + 37T^{2} \) |
| 41 | \( 1 - 7.88T + 41T^{2} \) |
| 43 | \( 1 + 0.975T + 43T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 - 1.47T + 59T^{2} \) |
| 61 | \( 1 + 4.24T + 61T^{2} \) |
| 67 | \( 1 - 6.74T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 + 5.02T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 - 4.35T + 89T^{2} \) |
| 97 | \( 1 + 3.75T + 97T^{2} \) |
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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
−9.218169936895908471074162013214, −8.287617189628156252583816964597, −6.92134943470379853560250713443, −6.46239491474346456039077817011, −5.97702098799257871085606062883, −4.98247156281963945267549344676, −4.22984552967165552012042898578, −3.52534545507026369945491233549, −2.60968081231717799704858326995, −0.983878420232591295657335447188,
0.983878420232591295657335447188, 2.60968081231717799704858326995, 3.52534545507026369945491233549, 4.22984552967165552012042898578, 4.98247156281963945267549344676, 5.97702098799257871085606062883, 6.46239491474346456039077817011, 6.92134943470379853560250713443, 8.287617189628156252583816964597, 9.218169936895908471074162013214
