L(s) = 1 | − 3-s + 5-s + 9-s − 11-s − 2·13-s − 15-s + 6·17-s − 8·19-s − 6·23-s + 25-s − 27-s + 6·29-s − 2·31-s + 33-s + 2·37-s + 2·39-s + 8·43-s + 45-s + 12·47-s − 6·51-s + 6·53-s − 55-s + 8·57-s − 6·59-s − 8·61-s − 2·65-s + 2·67-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 0.258·15-s + 1.45·17-s − 1.83·19-s − 1.25·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.359·31-s + 0.174·33-s + 0.328·37-s + 0.320·39-s + 1.21·43-s + 0.149·45-s + 1.75·47-s − 0.840·51-s + 0.824·53-s − 0.134·55-s + 1.05·57-s − 0.781·59-s − 1.02·61-s − 0.248·65-s + 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.505849866\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.505849866\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−14.99463205666567, −14.54613525406964, −13.91845656715472, −13.60179799064420, −12.65669287619881, −12.36577154681133, −12.15045560860605, −11.25136670407632, −10.60194709490337, −10.38257279673239, −9.787277963334880, −9.229532210168243, −8.492360064735371, −7.928781520160039, −7.407684784160954, −6.685381622147784, −6.074849845628143, −5.698095729903304, −5.059306199179439, −4.296471182346595, −3.879476625065096, −2.769178037637018, −2.283996813676282, −1.407219077667989, −0.4909082592640266,
0.4909082592640266, 1.407219077667989, 2.283996813676282, 2.769178037637018, 3.879476625065096, 4.296471182346595, 5.059306199179439, 5.698095729903304, 6.074849845628143, 6.685381622147784, 7.407684784160954, 7.928781520160039, 8.492360064735371, 9.229532210168243, 9.787277963334880, 10.38257279673239, 10.60194709490337, 11.25136670407632, 12.15045560860605, 12.36577154681133, 12.65669287619881, 13.60179799064420, 13.91845656715472, 14.54613525406964, 14.99463205666567