| L(s) = 1 | − 2·4-s + 4·7-s + 6·11-s − 13-s + 4·16-s + 6·17-s − 4·19-s + 3·23-s − 8·28-s + 3·29-s − 4·31-s − 2·37-s − 6·41-s + 7·43-s − 12·44-s + 9·49-s + 2·52-s − 9·53-s + 6·59-s − 61-s − 8·64-s − 14·67-s − 12·68-s + 6·71-s + 4·73-s + 8·76-s + 24·77-s + ⋯ |
| L(s) = 1 | − 4-s + 1.51·7-s + 1.80·11-s − 0.277·13-s + 16-s + 1.45·17-s − 0.917·19-s + 0.625·23-s − 1.51·28-s + 0.557·29-s − 0.718·31-s − 0.328·37-s − 0.937·41-s + 1.06·43-s − 1.80·44-s + 9/7·49-s + 0.277·52-s − 1.23·53-s + 0.781·59-s − 0.128·61-s − 64-s − 1.71·67-s − 1.45·68-s + 0.712·71-s + 0.468·73-s + 0.917·76-s + 2.73·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.050214354\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.050214354\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.801456054528029671345137936397, −8.109622323430795384540902637182, −7.44313533967700149106082663018, −6.43995160346575362242193439929, −5.50324850311908281715674980863, −4.79815352058635420780275090547, −4.15253391774556813403352263543, −3.35745100328651024034839394504, −1.76397611793551290887183908518, −0.989438760515621806827446532171,
0.989438760515621806827446532171, 1.76397611793551290887183908518, 3.35745100328651024034839394504, 4.15253391774556813403352263543, 4.79815352058635420780275090547, 5.50324850311908281715674980863, 6.43995160346575362242193439929, 7.44313533967700149106082663018, 8.109622323430795384540902637182, 8.801456054528029671345137936397
