L(s) = 1 | + 3-s − 5-s + 9-s − 5·13-s − 15-s + 2·17-s − 4·19-s − 4·25-s + 27-s − 5·29-s − 4·31-s − 10·37-s − 5·39-s − 5·41-s − 8·43-s − 45-s + 4·47-s − 7·49-s + 2·51-s + 11·53-s − 4·57-s + 4·59-s − 61-s + 5·65-s + 4·67-s + 12·71-s + 3·73-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.38·13-s − 0.258·15-s + 0.485·17-s − 0.917·19-s − 4/5·25-s + 0.192·27-s − 0.928·29-s − 0.718·31-s − 1.64·37-s − 0.800·39-s − 0.780·41-s − 1.21·43-s − 0.149·45-s + 0.583·47-s − 49-s + 0.280·51-s + 1.51·53-s − 0.529·57-s + 0.520·59-s − 0.128·61-s + 0.620·65-s + 0.488·67-s + 1.42·71-s + 0.351·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.227893037\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.227893037\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−15.24561067787523, −14.74387318899473, −14.52832206689048, −13.67071710915938, −13.32252502194307, −12.59369882224626, −12.11630772329132, −11.77841307721365, −10.90412380907866, −10.45447106695392, −9.740971577301442, −9.438106291707603, −8.637514069181490, −8.135338864329198, −7.649381750847242, −6.963473646043995, −6.638201661175068, −5.446898590901368, −5.210070451033466, −4.277842928069599, −3.725845267024700, −3.147860141109490, −2.171857220305130, −1.800903732951002, −0.3983715460080655,
0.3983715460080655, 1.800903732951002, 2.171857220305130, 3.147860141109490, 3.725845267024700, 4.277842928069599, 5.210070451033466, 5.446898590901368, 6.638201661175068, 6.963473646043995, 7.649381750847242, 8.135338864329198, 8.637514069181490, 9.438106291707603, 9.740971577301442, 10.45447106695392, 10.90412380907866, 11.77841307721365, 12.11630772329132, 12.59369882224626, 13.32252502194307, 13.67071710915938, 14.52832206689048, 14.74387318899473, 15.24561067787523