| L(s) = 1 | − 2·3-s − 5-s + 9-s − 11-s − 4·13-s + 2·15-s − 4·17-s − 4·19-s + 2·23-s + 25-s + 4·27-s − 6·29-s + 4·31-s + 2·33-s − 2·37-s + 8·39-s − 2·41-s − 4·43-s − 45-s − 2·47-s − 7·49-s + 8·51-s − 2·53-s + 55-s + 8·57-s + 4·59-s + 2·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s + 1/3·9-s − 0.301·11-s − 1.10·13-s + 0.516·15-s − 0.970·17-s − 0.917·19-s + 0.417·23-s + 1/5·25-s + 0.769·27-s − 1.11·29-s + 0.718·31-s + 0.348·33-s − 0.328·37-s + 1.28·39-s − 0.312·41-s − 0.609·43-s − 0.149·45-s − 0.291·47-s − 49-s + 1.12·51-s − 0.274·53-s + 0.134·55-s + 1.05·57-s + 0.520·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4200896126\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4200896126\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−8.497791386386919616288171097576, −7.79052154349256935583029917519, −6.83983272591179757834964398656, −6.47997955549173230932120492013, −5.43128456824258070736944110076, −4.88724721499874105361129108398, −4.17866984900068017022364306747, −2.99285732721847931363982763734, −1.95617779492073068947971146982, −0.38489024844937336821881157559,
0.38489024844937336821881157559, 1.95617779492073068947971146982, 2.99285732721847931363982763734, 4.17866984900068017022364306747, 4.88724721499874105361129108398, 5.43128456824258070736944110076, 6.47997955549173230932120492013, 6.83983272591179757834964398656, 7.79052154349256935583029917519, 8.497791386386919616288171097576
