L(s) = 1 | + 3-s − 2·4-s − 5-s − 7-s + 9-s + 6·11-s − 2·12-s − 15-s + 4·16-s − 4·19-s + 2·20-s − 21-s − 6·23-s + 25-s + 27-s + 2·28-s − 6·29-s + 5·31-s + 6·33-s + 35-s − 2·36-s + 2·37-s + 11·43-s − 12·44-s − 45-s + 6·47-s + 4·48-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 4-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.80·11-s − 0.577·12-s − 0.258·15-s + 16-s − 0.917·19-s + 0.447·20-s − 0.218·21-s − 1.25·23-s + 1/5·25-s + 0.192·27-s + 0.377·28-s − 1.11·29-s + 0.898·31-s + 1.04·33-s + 0.169·35-s − 1/3·36-s + 0.328·37-s + 1.67·43-s − 1.80·44-s − 0.149·45-s + 0.875·47-s + 0.577·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.558312655\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.558312655\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−9.008930910832740607560012084607, −8.257123602358388337442955194591, −7.57956281051642326006188534285, −6.55524582534573198925304635730, −5.92304072421045057762692872270, −4.63926054043737451433495067052, −3.95459008578551647729761482809, −3.56366223357914753165489698545, −2.11990586499282594585564805475, −0.78831515756259562175962538725,
0.78831515756259562175962538725, 2.11990586499282594585564805475, 3.56366223357914753165489698545, 3.95459008578551647729761482809, 4.63926054043737451433495067052, 5.92304072421045057762692872270, 6.55524582534573198925304635730, 7.57956281051642326006188534285, 8.257123602358388337442955194591, 9.008930910832740607560012084607