| L(s) = 1 | − 2·4-s − 7-s + 13-s + 4·16-s + 3·17-s + 19-s + 2·28-s − 4·31-s + 5·37-s + 3·41-s − 8·43-s − 12·47-s + 49-s − 2·52-s + 9·53-s + 12·59-s + 7·61-s − 8·64-s − 7·67-s − 6·68-s + 9·71-s − 5·73-s − 2·76-s + 4·79-s − 12·83-s − 91-s − 10·97-s + ⋯ |
| L(s) = 1 | − 4-s − 0.377·7-s + 0.277·13-s + 16-s + 0.727·17-s + 0.229·19-s + 0.377·28-s − 0.718·31-s + 0.821·37-s + 0.468·41-s − 1.21·43-s − 1.75·47-s + 1/7·49-s − 0.277·52-s + 1.23·53-s + 1.56·59-s + 0.896·61-s − 64-s − 0.855·67-s − 0.727·68-s + 1.06·71-s − 0.585·73-s − 0.229·76-s + 0.450·79-s − 1.31·83-s − 0.104·91-s − 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.534673312\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.534673312\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−13.06347705130671, −12.82161945820668, −12.27401347058156, −11.64802163035603, −11.36902334403565, −10.65668590610561, −10.06533017692355, −9.850717049332949, −9.394430282244407, −8.806999918931521, −8.370379698171316, −7.996670796326720, −7.367355026278820, −6.869445157691661, −6.278660004462125, −5.592841178058904, −5.414576993594894, −4.718571747497890, −4.192608147407951, −3.593003323856353, −3.278358786827039, −2.546567367128211, −1.726616210525862, −1.041902582679317, −0.4162330200609582,
0.4162330200609582, 1.041902582679317, 1.726616210525862, 2.546567367128211, 3.278358786827039, 3.593003323856353, 4.192608147407951, 4.718571747497890, 5.414576993594894, 5.592841178058904, 6.278660004462125, 6.869445157691661, 7.367355026278820, 7.996670796326720, 8.370379698171316, 8.806999918931521, 9.394430282244407, 9.850717049332949, 10.06533017692355, 10.65668590610561, 11.36902334403565, 11.64802163035603, 12.27401347058156, 12.82161945820668, 13.06347705130671
