L(s) = 1 | + 2-s + 4-s − 4·7-s + 8-s + 6·13-s − 4·14-s + 16-s + 2·17-s + 19-s + 8·23-s + 6·26-s − 4·28-s − 2·29-s − 8·31-s + 32-s + 2·34-s − 10·37-s + 38-s − 6·41-s + 8·43-s + 8·46-s + 9·49-s + 6·52-s − 2·53-s − 4·56-s − 2·58-s + 12·59-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.51·7-s + 0.353·8-s + 1.66·13-s − 1.06·14-s + 1/4·16-s + 0.485·17-s + 0.229·19-s + 1.66·23-s + 1.17·26-s − 0.755·28-s − 0.371·29-s − 1.43·31-s + 0.176·32-s + 0.342·34-s − 1.64·37-s + 0.162·38-s − 0.937·41-s + 1.21·43-s + 1.17·46-s + 9/7·49-s + 0.832·52-s − 0.274·53-s − 0.534·56-s − 0.262·58-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.978082640\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.978082640\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.46959967553563314326278142090, −6.94513470922233572982601272884, −6.38047285334029166191137174899, −5.66743340307203990237288817195, −5.19315616870853187572261703544, −3.99534258639292954724757910276, −3.44300859134331455355447906420, −3.06741600751537351573486320855, −1.84549386024609514652528685853, −0.76225969044242243742634102339,
0.76225969044242243742634102339, 1.84549386024609514652528685853, 3.06741600751537351573486320855, 3.44300859134331455355447906420, 3.99534258639292954724757910276, 5.19315616870853187572261703544, 5.66743340307203990237288817195, 6.38047285334029166191137174899, 6.94513470922233572982601272884, 7.46959967553563314326278142090