L(s) = 1 | + 3-s − 2·5-s + 9-s + 4·11-s − 2·13-s − 2·15-s − 2·17-s + 4·19-s + 8·23-s − 25-s + 27-s − 6·29-s + 8·31-s + 4·33-s − 6·37-s − 2·39-s + 6·41-s + 4·43-s − 2·45-s − 2·51-s + 2·53-s − 8·55-s + 4·57-s − 4·59-s − 2·61-s + 4·65-s − 4·67-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.516·15-s − 0.485·17-s + 0.917·19-s + 1.66·23-s − 1/5·25-s + 0.192·27-s − 1.11·29-s + 1.43·31-s + 0.696·33-s − 0.986·37-s − 0.320·39-s + 0.937·41-s + 0.609·43-s − 0.298·45-s − 0.280·51-s + 0.274·53-s − 1.07·55-s + 0.529·57-s − 0.520·59-s − 0.256·61-s + 0.496·65-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.305412200\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.305412200\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 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 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.56975735076810513912224977904, −7.22105516751355937830285941842, −6.54491697858797937973081677578, −5.63168120431032402488374725093, −4.69719753757653070842273737242, −4.18766143896287814919024894984, −3.41088997198904527337952221517, −2.81065807634666280216127599883, −1.69952852024005505825174252057, −0.72841599673746501614747483703,
0.72841599673746501614747483703, 1.69952852024005505825174252057, 2.81065807634666280216127599883, 3.41088997198904527337952221517, 4.18766143896287814919024894984, 4.69719753757653070842273737242, 5.63168120431032402488374725093, 6.54491697858797937973081677578, 7.22105516751355937830285941842, 7.56975735076810513912224977904