L(s) = 1 | + 4·7-s − 4·11-s + 2·13-s + 2·17-s + 4·19-s + 4·23-s + 2·29-s − 8·31-s − 6·37-s + 6·41-s + 8·43-s + 4·47-s + 9·49-s + 6·53-s + 4·59-s − 2·61-s − 8·67-s + 6·73-s − 16·77-s − 16·83-s + 6·89-s + 8·91-s + 14·97-s − 6·101-s − 4·103-s + 14·109-s + 18·113-s + ⋯ |
L(s) = 1 | + 1.51·7-s − 1.20·11-s + 0.554·13-s + 0.485·17-s + 0.917·19-s + 0.834·23-s + 0.371·29-s − 1.43·31-s − 0.986·37-s + 0.937·41-s + 1.21·43-s + 0.583·47-s + 9/7·49-s + 0.824·53-s + 0.520·59-s − 0.256·61-s − 0.977·67-s + 0.702·73-s − 1.82·77-s − 1.75·83-s + 0.635·89-s + 0.838·91-s + 1.42·97-s − 0.597·101-s − 0.394·103-s + 1.34·109-s + 1.69·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.085116964\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.085116964\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−9.100428369436491434363376978977, −8.469028469012419436497301616644, −7.61379691617992725579918349351, −7.25922965439850703725280828805, −5.74675556302869039795734872495, −5.29208484858772092309966421313, −4.45286390316500807643919809296, −3.31004324327388789633257437687, −2.19847701489412229516994368525, −1.05539838708270688643647945003,
1.05539838708270688643647945003, 2.19847701489412229516994368525, 3.31004324327388789633257437687, 4.45286390316500807643919809296, 5.29208484858772092309966421313, 5.74675556302869039795734872495, 7.25922965439850703725280828805, 7.61379691617992725579918349351, 8.469028469012419436497301616644, 9.100428369436491434363376978977