L(s) = 1 | + 3-s − 5-s − 7-s + 9-s − 4·11-s + 2·13-s − 15-s − 2·17-s − 21-s − 4·23-s + 25-s + 27-s + 2·29-s + 8·31-s − 4·33-s + 35-s + 10·37-s + 2·39-s + 2·41-s + 4·43-s − 45-s + 4·47-s + 49-s − 2·51-s + 10·53-s + 4·55-s + 4·59-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 0.258·15-s − 0.485·17-s − 0.218·21-s − 0.834·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 1.43·31-s − 0.696·33-s + 0.169·35-s + 1.64·37-s + 0.320·39-s + 0.312·41-s + 0.609·43-s − 0.149·45-s + 0.583·47-s + 1/7·49-s − 0.280·51-s + 1.37·53-s + 0.539·55-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.790051043\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.790051043\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 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
−8.452742804351938782384230250502, −7.994234340562527516798231668339, −7.29692828564724121688783820927, −6.39793835526705308468165926487, −5.66510656268215701234343947826, −4.57847845314384989370063128351, −3.96058883085238160252071042936, −2.92014264488954101491808603278, −2.30817877978589432011715052276, −0.75940214546438250197140839120,
0.75940214546438250197140839120, 2.30817877978589432011715052276, 2.92014264488954101491808603278, 3.96058883085238160252071042936, 4.57847845314384989370063128351, 5.66510656268215701234343947826, 6.39793835526705308468165926487, 7.29692828564724121688783820927, 7.994234340562527516798231668339, 8.452742804351938782384230250502