L(s) = 1 | + 2-s + 4-s − 5-s − 2·7-s + 8-s − 10-s + 2·11-s − 4·13-s − 2·14-s + 16-s − 6·17-s + 19-s − 20-s + 2·22-s − 8·23-s + 25-s − 4·26-s − 2·28-s + 6·29-s − 8·31-s + 32-s − 6·34-s + 2·35-s − 8·37-s + 38-s − 40-s + 12·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.755·7-s + 0.353·8-s − 0.316·10-s + 0.603·11-s − 1.10·13-s − 0.534·14-s + 1/4·16-s − 1.45·17-s + 0.229·19-s − 0.223·20-s + 0.426·22-s − 1.66·23-s + 1/5·25-s − 0.784·26-s − 0.377·28-s + 1.11·29-s − 1.43·31-s + 0.176·32-s − 1.02·34-s + 0.338·35-s − 1.31·37-s + 0.162·38-s − 0.158·40-s + 1.87·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 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
−9.025176372075570966398853964011, −8.006229175896152061430183432411, −7.12808468871614044827026471806, −6.54943624680207924278189450534, −5.70036907442361447785351628737, −4.60245607903835129854880943512, −3.99142551040147876305546488235, −2.98658523615693456704100119267, −1.96605921276206524911761642145, 0,
1.96605921276206524911761642145, 2.98658523615693456704100119267, 3.99142551040147876305546488235, 4.60245607903835129854880943512, 5.70036907442361447785351628737, 6.54943624680207924278189450534, 7.12808468871614044827026471806, 8.006229175896152061430183432411, 9.025176372075570966398853964011