L(s) = 1 | − 2.05·2-s − 3-s + 2.21·4-s − 0.0248·5-s + 2.05·6-s + 1.52·7-s − 0.449·8-s + 9-s + 0.0511·10-s + 11-s − 2.21·12-s − 2.02·13-s − 3.13·14-s + 0.0248·15-s − 3.51·16-s + 4.80·17-s − 2.05·18-s − 3.29·19-s − 0.0552·20-s − 1.52·21-s − 2.05·22-s − 1.24·23-s + 0.449·24-s − 4.99·25-s + 4.15·26-s − 27-s + 3.39·28-s + ⋯ |
L(s) = 1 | − 1.45·2-s − 0.577·3-s + 1.10·4-s − 0.0111·5-s + 0.838·6-s + 0.577·7-s − 0.158·8-s + 0.333·9-s + 0.0161·10-s + 0.301·11-s − 0.640·12-s − 0.561·13-s − 0.839·14-s + 0.00642·15-s − 0.878·16-s + 1.16·17-s − 0.484·18-s − 0.755·19-s − 0.0123·20-s − 0.333·21-s − 0.437·22-s − 0.260·23-s + 0.0916·24-s − 0.999·25-s + 0.815·26-s − 0.192·27-s + 0.640·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
good | 2 | \( 1 + 2.05T + 2T^{2} \) |
| 5 | \( 1 + 0.0248T + 5T^{2} \) |
| 7 | \( 1 - 1.52T + 7T^{2} \) |
| 13 | \( 1 + 2.02T + 13T^{2} \) |
| 17 | \( 1 - 4.80T + 17T^{2} \) |
| 19 | \( 1 + 3.29T + 19T^{2} \) |
| 23 | \( 1 + 1.24T + 23T^{2} \) |
| 29 | \( 1 + 8.92T + 29T^{2} \) |
| 31 | \( 1 - 5.62T + 31T^{2} \) |
| 37 | \( 1 + 6.88T + 37T^{2} \) |
| 41 | \( 1 - 8.93T + 41T^{2} \) |
| 43 | \( 1 + 6.34T + 43T^{2} \) |
| 47 | \( 1 - 4.19T + 47T^{2} \) |
| 53 | \( 1 + 3.56T + 53T^{2} \) |
| 59 | \( 1 - 1.96T + 59T^{2} \) |
| 67 | \( 1 + 2.12T + 67T^{2} \) |
| 71 | \( 1 - 8.06T + 71T^{2} \) |
| 73 | \( 1 - 0.798T + 73T^{2} \) |
| 79 | \( 1 + 0.406T + 79T^{2} \) |
| 83 | \( 1 + 1.36T + 83T^{2} \) |
| 89 | \( 1 + 8.42T + 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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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
−8.797936055434604462341479187467, −7.933568956414431081201072949468, −7.53712885267111789529147754655, −6.62486824985528285181210270251, −5.71360354927539045188660360520, −4.79649799692696371420756265426, −3.79392669043590819568676315355, −2.21320520023075297658826176159, −1.31881167459163369689341966559, 0,
1.31881167459163369689341966559, 2.21320520023075297658826176159, 3.79392669043590819568676315355, 4.79649799692696371420756265426, 5.71360354927539045188660360520, 6.62486824985528285181210270251, 7.53712885267111789529147754655, 7.933568956414431081201072949468, 8.797936055434604462341479187467