L(s) = 1 | − 3-s − 5-s + 9-s − 1.64·11-s + 2.64·13-s + 15-s + 1.64·17-s − 8.29·19-s + 1.64·23-s + 25-s − 27-s + 7.64·29-s + 4.29·31-s + 1.64·33-s + 0.645·37-s − 2.64·39-s − 4.93·41-s − 5.93·43-s − 45-s − 6·47-s − 1.64·51-s + 3.29·53-s + 1.64·55-s + 8.29·57-s − 10.9·59-s − 8·61-s − 2.64·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.333·9-s − 0.496·11-s + 0.733·13-s + 0.258·15-s + 0.399·17-s − 1.90·19-s + 0.343·23-s + 0.200·25-s − 0.192·27-s + 1.41·29-s + 0.770·31-s + 0.286·33-s + 0.106·37-s − 0.423·39-s − 0.771·41-s − 0.905·43-s − 0.149·45-s − 0.875·47-s − 0.230·51-s + 0.452·53-s + 0.221·55-s + 1.09·57-s − 1.42·59-s − 1.02·61-s − 0.328·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 1.64T + 11T^{2} \) |
| 13 | \( 1 - 2.64T + 13T^{2} \) |
| 17 | \( 1 - 1.64T + 17T^{2} \) |
| 19 | \( 1 + 8.29T + 19T^{2} \) |
| 23 | \( 1 - 1.64T + 23T^{2} \) |
| 29 | \( 1 - 7.64T + 29T^{2} \) |
| 31 | \( 1 - 4.29T + 31T^{2} \) |
| 37 | \( 1 - 0.645T + 37T^{2} \) |
| 41 | \( 1 + 4.93T + 41T^{2} \) |
| 43 | \( 1 + 5.93T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 - 3.29T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 - 0.645T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 - 2.29T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−8.307462494790344301334692836849, −7.75112764928530717455200414400, −6.47866242269807619472835415173, −6.43036861165827495316228777643, −5.17164068998328957742613280428, −4.55429118265446667781241671394, −3.66199299500958599632960753517, −2.63450416022290959496984609206, −1.34341653690519261556267413256, 0,
1.34341653690519261556267413256, 2.63450416022290959496984609206, 3.66199299500958599632960753517, 4.55429118265446667781241671394, 5.17164068998328957742613280428, 6.43036861165827495316228777643, 6.47866242269807619472835415173, 7.75112764928530717455200414400, 8.307462494790344301334692836849