L(s) = 1 | − 2-s + 4-s + 2·5-s − 8-s − 2·10-s + 11-s − 2·13-s + 16-s + 2·17-s + 2·20-s − 22-s + 8·23-s − 25-s + 2·26-s + 2·29-s + 8·31-s − 32-s − 2·34-s − 2·37-s − 2·40-s + 10·41-s + 4·43-s + 44-s − 8·46-s + 8·47-s + 50-s − 2·52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.353·8-s − 0.632·10-s + 0.301·11-s − 0.554·13-s + 1/4·16-s + 0.485·17-s + 0.447·20-s − 0.213·22-s + 1.66·23-s − 1/5·25-s + 0.392·26-s + 0.371·29-s + 1.43·31-s − 0.176·32-s − 0.342·34-s − 0.328·37-s − 0.316·40-s + 1.56·41-s + 0.609·43-s + 0.150·44-s − 1.17·46-s + 1.16·47-s + 0.141·50-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.966391127\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.966391127\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−7.58968718829955952946248212978, −7.16133368429204365756844004030, −6.27331078721880516383014073489, −5.86900440659026317386454609890, −4.99921866879504808571538824369, −4.28366336676484992028679824755, −3.06481182699252878563183531151, −2.57138081896173818686984856609, −1.57031058712854000015521935554, −0.78440603914936073147968083830,
0.78440603914936073147968083830, 1.57031058712854000015521935554, 2.57138081896173818686984856609, 3.06481182699252878563183531151, 4.28366336676484992028679824755, 4.99921866879504808571538824369, 5.86900440659026317386454609890, 6.27331078721880516383014073489, 7.16133368429204365756844004030, 7.58968718829955952946248212978