| L(s) = 1 | + 3-s + 9-s − 13-s − 2·17-s + 4·19-s + 27-s + 6·29-s + 2·37-s − 39-s + 6·41-s − 12·43-s − 4·47-s − 7·49-s − 2·51-s − 6·53-s + 4·57-s + 8·59-s − 2·61-s + 4·67-s + 12·71-s + 14·73-s + 81-s + 8·83-s + 6·87-s − 18·89-s + 6·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.277·13-s − 0.485·17-s + 0.917·19-s + 0.192·27-s + 1.11·29-s + 0.328·37-s − 0.160·39-s + 0.937·41-s − 1.82·43-s − 0.583·47-s − 49-s − 0.280·51-s − 0.824·53-s + 0.529·57-s + 1.04·59-s − 0.256·61-s + 0.488·67-s + 1.42·71-s + 1.63·73-s + 1/9·81-s + 0.878·83-s + 0.643·87-s − 1.90·89-s + 0.609·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.681311807\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.681311807\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−15.83318689842678, −15.52738418795665, −14.86029636876432, −14.22779488900974, −13.94810413050231, −13.20825795327557, −12.76163100332172, −12.13967086763265, −11.44258517489623, −11.04770095619127, −10.09957275406435, −9.794877003068180, −9.190476668968418, −8.444029104829780, −8.035118743378285, −7.361378357489470, −6.667401202248876, −6.188706753350870, −5.094577484550590, −4.807221045977956, −3.834428360130568, −3.219349507905487, −2.498047915129968, −1.698320682984274, −0.6918101598464083,
0.6918101598464083, 1.698320682984274, 2.498047915129968, 3.219349507905487, 3.834428360130568, 4.807221045977956, 5.094577484550590, 6.188706753350870, 6.667401202248876, 7.361378357489470, 8.035118743378285, 8.444029104829780, 9.190476668968418, 9.794877003068180, 10.09957275406435, 11.04770095619127, 11.44258517489623, 12.13967086763265, 12.76163100332172, 13.20825795327557, 13.94810413050231, 14.22779488900974, 14.86029636876432, 15.52738418795665, 15.83318689842678
