L(s) = 1 | + 5-s + 2·7-s + 2·11-s − 2·17-s + 4·19-s + 25-s + 2·29-s + 8·31-s + 2·35-s − 4·37-s − 8·41-s + 8·43-s − 8·47-s − 3·49-s + 10·53-s + 2·55-s − 6·59-s + 2·61-s + 12·67-s + 12·71-s − 2·73-s + 4·77-s + 8·79-s − 4·83-s − 2·85-s − 12·89-s + 4·95-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.755·7-s + 0.603·11-s − 0.485·17-s + 0.917·19-s + 1/5·25-s + 0.371·29-s + 1.43·31-s + 0.338·35-s − 0.657·37-s − 1.24·41-s + 1.21·43-s − 1.16·47-s − 3/7·49-s + 1.37·53-s + 0.269·55-s − 0.781·59-s + 0.256·61-s + 1.46·67-s + 1.42·71-s − 0.234·73-s + 0.455·77-s + 0.900·79-s − 0.439·83-s − 0.216·85-s − 1.27·89-s + 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.116841543\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.116841543\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−9.567162894678014522729592248666, −8.687678118107818094251078951166, −8.044486700043922892061139958060, −7.02269464275919322402346076699, −6.31857937347981786877789666682, −5.28368610102458007670837638108, −4.59174388082872419767425141006, −3.46292942161717526500805424747, −2.25866714014751539772152464560, −1.14198203052138198817329951084,
1.14198203052138198817329951084, 2.25866714014751539772152464560, 3.46292942161717526500805424747, 4.59174388082872419767425141006, 5.28368610102458007670837638108, 6.31857937347981786877789666682, 7.02269464275919322402346076699, 8.044486700043922892061139958060, 8.687678118107818094251078951166, 9.567162894678014522729592248666