| L(s) = 1 | − 4·5-s − 4·7-s − 4·11-s − 13-s − 8·23-s + 11·25-s + 8·29-s − 4·31-s + 16·35-s + 6·37-s − 12·41-s + 8·43-s − 4·47-s + 9·49-s + 16·55-s + 4·59-s − 2·61-s + 4·65-s + 8·67-s − 4·71-s − 10·73-s + 16·77-s + 4·79-s + 12·83-s + 12·89-s + 4·91-s + 14·97-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 1.51·7-s − 1.20·11-s − 0.277·13-s − 1.66·23-s + 11/5·25-s + 1.48·29-s − 0.718·31-s + 2.70·35-s + 0.986·37-s − 1.87·41-s + 1.21·43-s − 0.583·47-s + 9/7·49-s + 2.15·55-s + 0.520·59-s − 0.256·61-s + 0.496·65-s + 0.977·67-s − 0.474·71-s − 1.17·73-s + 1.82·77-s + 0.450·79-s + 1.31·83-s + 1.27·89-s + 0.419·91-s + 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3822763492\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3822763492\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−9.193654244805756746325292550091, −8.201733704151596185987903064087, −7.79060854775042365605164468705, −6.94787299886291680521490460261, −6.19578640602053014565524878691, −5.05645028216135290095057077475, −4.10665661851761688347162074488, −3.38922081885229144696621278738, −2.58928593205915212490307519815, −0.38897534472433013470827993329,
0.38897534472433013470827993329, 2.58928593205915212490307519815, 3.38922081885229144696621278738, 4.10665661851761688347162074488, 5.05645028216135290095057077475, 6.19578640602053014565524878691, 6.94787299886291680521490460261, 7.79060854775042365605164468705, 8.201733704151596185987903064087, 9.193654244805756746325292550091
