L(s) = 1 | − 5-s − 7-s − 3.12·11-s + 2·13-s + 3.12·17-s + 1.12·19-s + 3.12·23-s + 25-s − 5.12·29-s − 6.24·31-s + 35-s + 7.12·37-s − 8.24·41-s + 1.12·43-s + 1.12·47-s + 49-s + 1.12·53-s + 3.12·55-s − 4·59-s + 11.1·61-s − 2·65-s + 1.12·67-s − 6·71-s − 4.24·73-s + 3.12·77-s − 8·79-s − 3.12·85-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.377·7-s − 0.941·11-s + 0.554·13-s + 0.757·17-s + 0.257·19-s + 0.651·23-s + 0.200·25-s − 0.951·29-s − 1.12·31-s + 0.169·35-s + 1.17·37-s − 1.28·41-s + 0.171·43-s + 0.163·47-s + 0.142·49-s + 0.154·53-s + 0.421·55-s − 0.520·59-s + 1.42·61-s − 0.248·65-s + 0.137·67-s − 0.712·71-s − 0.496·73-s + 0.355·77-s − 0.900·79-s − 0.338·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + 3.12T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 3.12T + 17T^{2} \) |
| 19 | \( 1 - 1.12T + 19T^{2} \) |
| 23 | \( 1 - 3.12T + 23T^{2} \) |
| 29 | \( 1 + 5.12T + 29T^{2} \) |
| 31 | \( 1 + 6.24T + 31T^{2} \) |
| 37 | \( 1 - 7.12T + 37T^{2} \) |
| 41 | \( 1 + 8.24T + 41T^{2} \) |
| 43 | \( 1 - 1.12T + 43T^{2} \) |
| 47 | \( 1 - 1.12T + 47T^{2} \) |
| 53 | \( 1 - 1.12T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 - 1.12T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 4.24T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 4.24T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−7.74286288587011462319866655745, −7.34984851742976780740715696420, −6.45105325261502787659884143939, −5.59599324154609570957824234089, −5.08203824925009231218377799139, −3.98971777072567225046172167283, −3.35332038933079003920831542906, −2.52296608373889989696082746114, −1.27623074416991050030758372223, 0,
1.27623074416991050030758372223, 2.52296608373889989696082746114, 3.35332038933079003920831542906, 3.98971777072567225046172167283, 5.08203824925009231218377799139, 5.59599324154609570957824234089, 6.45105325261502787659884143939, 7.34984851742976780740715696420, 7.74286288587011462319866655745