| L(s) = 1 | − 2.67·5-s − 4.10·7-s − 11-s − 3.81·13-s − 5.91·17-s − 5.81·19-s + 3.67·23-s + 2.14·25-s + 8.38·29-s + 6.96·31-s + 10.9·35-s + 6.52·37-s + 7.24·41-s − 10.7·43-s − 0.143·47-s + 9.81·49-s − 4.48·53-s + 2.67·55-s + 11.8·59-s − 1.61·61-s + 10.2·65-s − 6.96·67-s − 0.489·71-s − 8.48·73-s + 4.10·77-s − 15.9·79-s − 5.16·83-s + ⋯ |
| L(s) = 1 | − 1.19·5-s − 1.54·7-s − 0.301·11-s − 1.05·13-s − 1.43·17-s − 1.33·19-s + 0.765·23-s + 0.428·25-s + 1.55·29-s + 1.25·31-s + 1.85·35-s + 1.07·37-s + 1.13·41-s − 1.64·43-s − 0.0210·47-s + 1.40·49-s − 0.616·53-s + 0.360·55-s + 1.54·59-s − 0.206·61-s + 1.26·65-s − 0.850·67-s − 0.0581·71-s − 0.993·73-s + 0.467·77-s − 1.79·79-s − 0.566·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4928388224\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4928388224\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 2.67T + 5T^{2} \) |
| 7 | \( 1 + 4.10T + 7T^{2} \) |
| 13 | \( 1 + 3.81T + 13T^{2} \) |
| 17 | \( 1 + 5.91T + 17T^{2} \) |
| 19 | \( 1 + 5.81T + 19T^{2} \) |
| 23 | \( 1 - 3.67T + 23T^{2} \) |
| 29 | \( 1 - 8.38T + 29T^{2} \) |
| 31 | \( 1 - 6.96T + 31T^{2} \) |
| 37 | \( 1 - 6.52T + 37T^{2} \) |
| 41 | \( 1 - 7.24T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 + 0.143T + 47T^{2} \) |
| 53 | \( 1 + 4.48T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 + 1.61T + 61T^{2} \) |
| 67 | \( 1 + 6.96T + 67T^{2} \) |
| 71 | \( 1 + 0.489T + 71T^{2} \) |
| 73 | \( 1 + 8.48T + 73T^{2} \) |
| 79 | \( 1 + 15.9T + 79T^{2} \) |
| 83 | \( 1 + 5.16T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 7.38T + 97T^{2} \) |
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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
−8.828519659929503097262161799305, −8.312033199819185786688168356165, −7.30990854222680139761277628737, −6.71415192581902656936847399690, −6.11818522472653199760600187109, −4.65484656129505529893729940562, −4.28586932171741654735201054122, −3.10124369080828968200914530076, −2.49875142805317180810603258817, −0.41837354985317816800260393465,
0.41837354985317816800260393465, 2.49875142805317180810603258817, 3.10124369080828968200914530076, 4.28586932171741654735201054122, 4.65484656129505529893729940562, 6.11818522472653199760600187109, 6.71415192581902656936847399690, 7.30990854222680139761277628737, 8.312033199819185786688168356165, 8.828519659929503097262161799305
