| L(s) = 1 | + 3-s + 5-s + 9-s − 6·13-s + 15-s + 2·17-s + 6·23-s + 25-s + 27-s − 6·29-s + 2·31-s + 10·37-s − 6·39-s − 8·41-s + 8·43-s + 45-s − 4·47-s + 2·51-s + 6·53-s + 6·59-s − 8·61-s − 6·65-s + 14·67-s + 6·69-s − 8·71-s − 2·73-s + 75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.66·13-s + 0.258·15-s + 0.485·17-s + 1.25·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 0.359·31-s + 1.64·37-s − 0.960·39-s − 1.24·41-s + 1.21·43-s + 0.149·45-s − 0.583·47-s + 0.280·51-s + 0.824·53-s + 0.781·59-s − 1.02·61-s − 0.744·65-s + 1.71·67-s + 0.722·69-s − 0.949·71-s − 0.234·73-s + 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 355740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 355740 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.808298295\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.808298295\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.53171281252771, −12.15398195719531, −11.74482977521050, −11.06641370301491, −10.74244828640816, −10.12674166715222, −9.709901269771899, −9.401131495272866, −9.059932500971291, −8.404042976470854, −7.867426313101095, −7.537441845715812, −7.043055680018553, −6.647567648051350, −5.976268097150654, −5.472447260301546, −4.894590592993705, −4.689277070724935, −3.885230965421093, −3.396408288824638, −2.775782018846701, −2.368286314246553, −1.912378548244925, −1.089547580907269, −0.5153272505605460,
0.5153272505605460, 1.089547580907269, 1.912378548244925, 2.368286314246553, 2.775782018846701, 3.396408288824638, 3.885230965421093, 4.689277070724935, 4.894590592993705, 5.472447260301546, 5.976268097150654, 6.647567648051350, 7.043055680018553, 7.537441845715812, 7.867426313101095, 8.404042976470854, 9.059932500971291, 9.401131495272866, 9.709901269771899, 10.12674166715222, 10.74244828640816, 11.06641370301491, 11.74482977521050, 12.15398195719531, 12.53171281252771
