| L(s) = 1 | − 4·7-s − 2·11-s − 13-s − 2·19-s − 2·23-s − 10·29-s − 4·31-s + 6·37-s + 6·41-s − 8·43-s + 12·47-s + 9·49-s − 14·53-s + 6·59-s + 2·61-s + 4·67-s − 14·73-s + 8·77-s − 12·83-s − 6·89-s + 4·91-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 0.603·11-s − 0.277·13-s − 0.458·19-s − 0.417·23-s − 1.85·29-s − 0.718·31-s + 0.986·37-s + 0.937·41-s − 1.21·43-s + 1.75·47-s + 9/7·49-s − 1.92·53-s + 0.781·59-s + 0.256·61-s + 0.488·67-s − 1.63·73-s + 0.911·77-s − 1.31·83-s − 0.635·89-s + 0.419·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.41721212411232, −13.65930639260751, −13.20612180268292, −12.86866048727535, −12.53963510402806, −11.98312791309117, −11.18172462928541, −10.98498496643519, −10.24278187793973, −9.819390891087396, −9.429783600715098, −8.983110133351907, −8.304882841287085, −7.701947343667684, −7.228450443865199, −6.766375089020102, −6.060563659985146, −5.748528187861205, −5.194440366177467, −4.293610119358355, −3.904885474525755, −3.249865719744823, −2.669498283428417, −2.142447708050617, −1.240460830894593, 0, 0,
1.240460830894593, 2.142447708050617, 2.669498283428417, 3.249865719744823, 3.904885474525755, 4.293610119358355, 5.194440366177467, 5.748528187861205, 6.060563659985146, 6.766375089020102, 7.228450443865199, 7.701947343667684, 8.304882841287085, 8.983110133351907, 9.429783600715098, 9.819390891087396, 10.24278187793973, 10.98498496643519, 11.18172462928541, 11.98312791309117, 12.53963510402806, 12.86866048727535, 13.20612180268292, 13.65930639260751, 14.41721212411232
