L(s) = 1 | + 3·5-s + 7-s + 3·13-s + 4·17-s − 19-s − 4·23-s + 4·25-s − 3·29-s − 2·31-s + 3·35-s − 3·37-s − 4·43-s − 3·47-s + 49-s + 10·53-s − 3·59-s + 2·61-s + 9·65-s + 9·67-s + 16·71-s + 5·73-s + 6·79-s + 16·83-s + 12·85-s + 6·89-s + 3·91-s − 3·95-s + ⋯ |
L(s) = 1 | + 1.34·5-s + 0.377·7-s + 0.832·13-s + 0.970·17-s − 0.229·19-s − 0.834·23-s + 4/5·25-s − 0.557·29-s − 0.359·31-s + 0.507·35-s − 0.493·37-s − 0.609·43-s − 0.437·47-s + 1/7·49-s + 1.37·53-s − 0.390·59-s + 0.256·61-s + 1.11·65-s + 1.09·67-s + 1.89·71-s + 0.585·73-s + 0.675·79-s + 1.75·83-s + 1.30·85-s + 0.635·89-s + 0.314·91-s − 0.307·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.919046992\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.919046992\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−14.23284614764899, −13.81254050771153, −13.32026244548148, −12.96839393318323, −12.20870905663648, −11.89287016295946, −11.09439156109141, −10.73808668132572, −10.11461212112478, −9.779288286051638, −9.192040093939540, −8.709651860314540, −8.028561442196803, −7.690570094402940, −6.752578203727168, −6.399269805927677, −5.794347976761981, −5.308208201938780, −4.927673636761413, −3.770069000883079, −3.657792836287143, −2.569492004676482, −2.007343265899180, −1.487712278654654, −0.6798595822981586,
0.6798595822981586, 1.487712278654654, 2.007343265899180, 2.569492004676482, 3.657792836287143, 3.770069000883079, 4.927673636761413, 5.308208201938780, 5.794347976761981, 6.399269805927677, 6.752578203727168, 7.690570094402940, 8.028561442196803, 8.709651860314540, 9.192040093939540, 9.779288286051638, 10.11461212112478, 10.73808668132572, 11.09439156109141, 11.89287016295946, 12.20870905663648, 12.96839393318323, 13.32026244548148, 13.81254050771153, 14.23284614764899